AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER――FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

GU Xu-zan; ZHANG Bing; WANG Ming-huan

Abstract:

In this paper, the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion. Here we introduce a cubic spline numerical model (Spline Model for short), which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh. A new algorithm of "fitting cubic spline―time step integration―fitting cubic spline―……" is developed to determine their first- and 2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model. And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model's mathematical foundation of numerical analysis. It is pointed out that the Spline Model has mathematical laws of "convergence" of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives. The "optimality" of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions. In addition, a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field. Besides, the slopes and curvatures of a central difference are identified respectively, with a smoothing coefficient of 1/3, three-point smoothing of that of a cubic spline. Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline, respectively. Furthermore, a global simulation case of adiabatic, non-frictional and "incompressible" model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model, whose initial condition comes from the NCEP reanalysis data, along with quasi-uniform latitude-longitude grids and the so-called "shallow atmosphere" Navier-Stokes primitive equations in the spherical coordinates. The Spline Model, which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme, provides an initial ideal case of global atmospheric circulation. In addition, considering the essentially non-linear atmospheric motions, the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

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GU Xu-zan, ZHANG Bing, WANG Ming-huan. AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER――FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS [J]. Journal of Tropical Meteorology, 2013, 19(4): 388-396.

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AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER――FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

Hubei Key Laboratory for Heavy Rain Monitoring and Warning Research, Institute of Heavy Rain, China Meteorological Administration, Wuhan 430074 China

Received Date: 2012-10-19

Rev Recd Date: 2013-10-15

Abstract: In this paper, the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion. Here we introduce a cubic spline numerical model (Spline Model for short), which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh. A new algorithm of "fitting cubic spline―time step integration―fitting cubic spline―……" is developed to determine their first- and 2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model. And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model's mathematical foundation of numerical analysis. It is pointed out that the Spline Model has mathematical laws of "convergence" of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives. The "optimality" of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions. In addition, a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field. Besides, the slopes and curvatures of a central difference are identified respectively, with a smoothing coefficient of 1/3, three-point smoothing of that of a cubic spline. Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline, respectively. Furthermore, a global simulation case of adiabatic, non-frictional and "incompressible" model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model, whose initial condition comes from the NCEP reanalysis data, along with quasi-uniform latitude-longitude grids and the so-called "shallow atmosphere" Navier-Stokes primitive equations in the spherical coordinates. The Spline Model, which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme, provides an initial ideal case of global atmospheric circulation. In addition, considering the essentially non-linear atmospheric motions, the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

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[1]	XI Mei-cheng. Numerical Analysis Methods.[M]. Hefei: China Science and Technology University Press, 2003: 1-54.
[2]	LI Jian-ping. Tutorial Program of Computer Graphics Principle.[M]. Chengdu: Electronic Science and Technology University Press, 1988: 76-82; 104-123.
[3]	SUN Jia-guang. Computer Graphics.[M]. Beijing: Tsinghua University Press, 1998: 286-354.
[4]	ZHANG Quan-huo, ZHANG Jian-da. Computer Graphics.[M]. Beijing: Mechanical Industry Press, 2003: 60-61, 83-98.
[5]	FERGUSION J C. Multivariable curve interpolation.[J]. J. Assoc. Comput. Mach., 1964, 2: 221-228.
[6]	ROBERT A A. Semi-Lagrangian and semi-implicit numerical integration scheme for the primitive meteorological equations[J]. J. Meteor. Soc. Japan, 1982, 60: 319-325.
[7]	ROBERT A A, YEE T L, RITCHIE H A. Semi-Lagrangian and semi-implicit numerical integration scheme for multilevel atmospheric models.[J]. Mon. Wea. Rev., 1985, 113: 388-394.
[8]	BATES J R, SEMAZZI F H M, HIGGINS R W, et al. Integration of the shallow water equations on the sphere using a vector semi-Lagrangian scheme with a multigrid solver.[J]. Mon. Wea. Rev., 1990, 118: 615-627.
[9]	QIAN Jian-hua, SEMAZZI F H M, JEFFREY S S. A global non-hydrostatic semi-Lagrangian, atmospheric model with orography.[J]. Mon. Wea. Rev., 1998, 126: 747-771.
[10]	PURSER R J, LESLIE L M. An efficient interpolation procedure for high-order three-dimensional semi- Lagrangian models.[J]. Mon. Wea. Rev., 1991, 119: 2492-2498.
[11]	NOIR R, COTE J, STANIFORTH A. Monotonic cascade interpolation for semi-Lagrangian advection.[J]. Quart. J. R. Met. Soc., 1999, 125: 197-212.
[12]	LAYTON A T. Cubic spline collocation method for the shallow water equations on the sphere.[J]. Comput. Phys., 2002, 179: 578-592.
[13]	GU Xu-zan, ZHANG Bing. Introduction of the bicubic numerical model.[J]. Meteor. Sci. Techn., 2006, 34(4): 353-357.
[14]	GU Xu-zan, ZHANG Bing. Design of a global Bicubic Numerical Model (BiNM) and its simulation casesⅠ: Design of dynamic core for BiNM.[J]. Plateau Meteor., 2008, 27(3): 474-480.
[15]	GU Xu-zan, ZHANG Bing. Design of a global Bicubic Numerical Model (BiNM) and its simulation casesⅡ: Simulation Case Analysis.[J]. Plateau Meteor., 2008, 27(3): 481-490.
[16]	GU Xu-zan. The mathematical consistency of quasi-Lagrangian and Eulerian integration schemes with tilting cubic interpolation functions.[J]. Plateau Meteor., 2010, 29(3): 655-661.
[17]	GU Xu-zan. A new quasi-Lagrangian Integration scheme with fitting cubic spline functions.[J]. Acta Meteor. Sinica, 2011, 69(3): 440-446.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER――FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

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