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Zhong Qing (Institute of Atmospheric Physics, Chinese Academy of Sciences,Beijing,100029)  
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.
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