三次样条函数（样条格式）为二阶可导非线性格式，但样条格式线性部分是二阶导数中央差。本文在简谐波真解条件下，推导证明二阶导数中央差比一阶导数中央差的空间截断误差以及相速和群速误差均减少一倍。借鉴谱模式动力框架核心思想，高斯网格二维谱变换半隐式-半拉格朗日积分方案，研究准均匀经纬网格样条格式变换显式-准拉格朗日积分方案。引入原始大气运动方程，推导样条格式二阶时空离散准拉格朗日预报方程通式，得出静力守恒气压、气温预报方程，在经纬网格基础上，设计两种基本准均匀经纬网格，通过对压、温、湿、风及广义牛顿力（加速度）场做“经纬网格-准均匀经纬网格”三次样条函数变换，求得“水平双三次曲面+垂直三次样条”拟合上游点三次运动路径，用“匀加速”变率预报风场，进而求得一个时间步长平均“静力平流”三维位移散度场，并用它预报气压场增压和气温场绝热增温，从而实现全球静力质量守恒经纬网格三次样条函数变换显式-准拉格朗日积分方案，经初步积分试验，证明上述动力框架是可行的。
The spline format is a no-linear, second-order derivative one, its linear segment is that of the second-order central difference. In this paper, we give a derivation proof of that the space truncation error, phase velocity and group velocity errors of the second-order center differential is halved that of the first-order center differential under a hypothesis of genuine solution of simple harmonic wave. So, we draw lessons from the idea of the dynamic core of spectral model, the semi-implic semi-Lagrangian integration scheme with 2D spectral spherical harmonic function transform on the Gussion grid, to introduce a new explicit quasi-Lagrangian integration scheme with cubic spline function transform on guasi-uniform latitude-longitude grid (called "spline model"). Adopting original atmospheric equations of motion, which includes that in the North Pole and the Sorth Pole, a general forecast equation of spline format of space-time second-order differential remainder is derived, then obtain the hydrostatic pressure and temperature forecast eqations of conservation of the atmospheric mass. Based on uniform latitude-longitude grid, we harmonize two quasi-uniform ones, which must be quasi-uniform latitude space, and on which cubic spline function transformation (transformation=fitting+interpolation) must be done for variables of pressure, temperature, moisture, winds and general Newtonian force acting to unit air mass on rotating earth (acceleration), which made all of them second-order derivative, to solve the track of an upstream point, but the upstream air parcel goes alone just "cubic path" of fitting their slopes, curvatures and torsions of the variable fields to "bicubic surface in horizontal + cubic spline in vertical". It is with a path of uniform acceleration motion to forecast wind field, and with fitting splines to the paths of the 3D hydrostatic advection and getting its implicit average divergence in one time step, to forecast increments of pressure and temperature fields in the adiabatic process. We give two integration cases that testify to the dynamic core of global spline model.
[1]Layton A T. 2002. Cubic spline collocation method for the shallow water equations on the sphere[J]. Journal of Computational Physics, 179(2):578-592.
[2]Bubnová R, Hello G, Bénard P, et al. 1995. Integration of the fully elastic equations cast in the hydrostatic pressure terrain-following coordinate in the framework of the ARPEGE/Aladin NWP system[J]. Mon Wea Rev, 123(2):515-535.
[3]C?té J, Gravel S, Méthot A, et al. 1998a. The operational CMC-MRB global environmental multiscale (GEM) model. Part Ⅰ:Design considerations and formulation[J]. Mon Wea Rev, 126(6):1373-1395.
[4]C?té J, Desmarais J G, Gravel S, et al. 1998b. The operational CMC|MRB global environmental multiscale (GEM) model. Part Ⅱ:results[J]. Mon Wea Rev, 126(6):1397-1418.
[5]Cullen M J P. 1990. A test of a semi-implicit integration technique for a fully compressible non-hydrostatic model[J]. Quart J Roy Meteor Soc, 116(495):1253-1258.
[6]Cullen M J P, Davies T, Mawson M H, et al. 1997. An overview of numerical methods for the next generation UK NWP and climate model[J]. Atmos-Ocean, 35(supp1):425-444.
[7]Ferguson J. 1964. Multivariable curve interpolation[J]. Journal of the ACM (JACM), 11(2):221-228.
[8]Gal-Chen T, Somerville R C J. 1975. On the use of a coordinate transformation for the solution of the Navier-Stokes equations[J]. Journal of Computational Physics, 17(2):209-228.
[9]Li X, Chen D, Peng X, et al. 2006. Implementation of the semi-Lagrangian advection scheme on a quasi-uniform overset grid on a sphere[J]. Adv Atmos Sci, 23(5):792-801.
[10]McDonald A, Bates J R. 1989. Semi-Lagrangian integration of a gridpoint shallow water model on the sphere[J]. Mon Wea Rev, 117(1):130-137.
[11]Ritchie H, Beaudoin C. 1994. Approximations and sensitivity experiments with a baroclinic semi-Lagrangian spectral model[J]. Mon Wea Rev, 122(10):2391-2399.
[12]Chen Dehui, Hu Zhijing, Xu Dahai, et al. 2004. Research on the system of atmospheric numerical prediction model (CAMS)[M]. Beijing:China Meteorological Press, 190.<br/>陈德辉, 胡志晋, 徐大海, 等. 2004. CAMS大气数值预报模式系统研究[M].北京:气象出版社, 190.
[13]Gu Xuzan. 2003. Compution and characteristic of a model atmospheric reference state[J]. Plateau Meteor, 22(6):608-612.<br/>辜旭赞. 2003.一模式大气参考状态的科学计算与特征分析[J].高原气象. 22(6):608-612.
[14]Gu Xuzan. 2010. The mathematical consistency of quasi-Lagrangian and Eulerian time integratifmn schemes with fitting ciibic interpolation functifmn[J]. Plateau Meteor, 29(3):655-661.<br/>辜旭赞. 2010. "准拉格朗日法"和"欧拉法"数学一致性与三次插值函数算法时间积分方案[J].高原气象, 29(3):655-661.
[15]Gu Xuzan. 2011. A new quasi-Lagrangian time integration scheme with the interpolation of fitting bicubic surface[J]. Acta Meteor Sinica, 69(3):440-446.<br/>辜旭赞. 2011.一种"准拉格朗日法"和"欧拉法"统一算法时间积分方案[J].气象学报, 69(3):440-446.
[16]Li Jianghao, Peng Xindong. 2013. Analysis of computational performance of conservative constraint on the Yin-Yang Grid[J]. Chinese J Atmos Sci, 37(4):852-862.<br/>李江浩, 彭新东. 2013.阴阳网格上质量守恒计算性能分析[J].大气科学, 37(4):852-862.
[17]Liao Dongxian. 1999. Design of atmospheric numerical model[M]. Beijing:China Meteorological Press, 291.<br/>廖洞贤. 1999.大气数值模式的设计[M].北京:气象出版社, 291.
[18]Lü Meizhong, Ouyang Ziji, Zhai Zihang. 1983. The semi-implicit spline scheme of the barotropic primitive equation[J]. J Meteor Sci, 4(2):1-11.<br/>吕美仲, 欧阳子济, 翟子航. 1983.正压原始方程半隐式样条格式[J].气象科学, 4(2):1-11.
[19]Xi Meicheng. 1995. Numerical analysis method[M]. Hefei:University of Science & Technology China Press, 377.<br/>奚梅成. 1995.数值分析方法[M].合肥:中国科学技术大学出版社, 377.
[20]Xue Jishan, Chen Dehui. 2008. Scientific design and application of the numerical prediction system (GRAPES)[M]. Beijing:Science Press, 383.<br/>薛纪善, 陈德辉. 2008.数值预报系统GRAPES的科学设计与应用[M].北京:科学出版社, 383.
[21]Zhang Yulin, Wu Huizhan, Wang Xiaolin. 1986. Numerical weather Prediction[M]. Beijing:Science Press, 472.<br/>张玉玲, 吴辉碇, 王晓林. 1986.数值天气预报[M].北京:科学出版社, 472.
[22]Zhou Yi, Liu Yudi, Gui Qijun, et al. 2002. Modern numerical weather prediction[M]. Beijing:China Meteorological Press, 220.<br/>周毅, 刘宇迪, 桂祁军, 等. 2002.现代数值天气预报[M].北京:气象出版社, 220.
