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

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 schemes.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-implicit model for baroclinic primitive equations is currently used in many countries in the world for operational weather forecast and numerical simulations 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 formulation problem remains 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 integration. The experiment of thirteen FGGE data 30-day numerical integration indicates that the new type of total energy conserving semi-implicit fidelity scheme can surely modify the systematic deviation of energy and mass conserving of the traditional scheme.It should be particularly noted that,under the experiment 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 root-mean-square(RMS)error at the end of second week of the integration and exceed 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 integration,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.

Elimination of Computational Systematic Errors and Improvements of Weather and Climate System Models in Relation to Baroclinic Primitive Equations

The design of a total energy conserving semi-implicit scheme for the multiple-level baroclinic primitive equation has remained an unsolved problem for a long time. In this work, however, we follow an energy perfect conserving semi-implicit scheme of a European Centre for Medium-Range Weather Forecasts (ECMWF) type sigma-coordinate primitive equation which has recently successfully formulated. Some real-data contrast tests between the model of the new conserving scheme and that of the ECMWF-type of global spectral semi-implicit scheme show that the RMS error of the averaged forecast Height at 850 hPa can be clearly improved after the first integral week. The reduction also reaches 50 percent by the 30th day. Further contrast tests demonstrate that the RMS error of the monthly mean height in the middle and lower troposphere also be largely reduced, and some well-known systematical defects can be greatly improved. More detailed analysis reveals that part of the positive contributions comes from improvements of the extra-long wave components. This indicates that a remarkable improvement of the model climate drift level can be achieved by the actual realizing of a conserving time-difference scheme, which thereby eliminates a corresponding computational systematic error source/sink found in the currently-used traditional type of weather and climate system models in relation to the baroclinic primitive equations.

LONG VALID TIME ENERGY PERFECT CONSERVATIVE FIDELITY SPECTRAL SCHEMES OF BAROTROPIC PRIMITIVE EQUATIONS

In accordance with a new compensation principle of discrete computations,the traditional meteo- rological global (pseudo-) spectral schemes of barotropic primitive equation (s) are transformed into perfect energy conservative fidelity schemes,thus resolving the problems of both nonlinear computa- tional instability and incomplete energy conservation,and raising the computational efficiency of the traditional schemes. As the numerical tests of the new schemes demonstrate,in solving the problem of energy conser- vation in operational computations,the new schemes can eliminate the (nonlinear) computational in- stability and,to some extent even the (nonlinear) computational diverging as found in the traditional schemes,Further contrasts between new and traditional schemes also indicate that,in discrete opera- tional computations,the new scheme in the case of nondivergence is capable of prolonging the valid in- tegral time of the corresponding traditional scheme,and eliminating certain kind of systematical com- putational“climate drift”,meanwhile increasing its computational accuracy and reducing its amount of computation.The working principle of this paper is also applicable to the problem concerning baroclin- ic primitive equations.