PROBLEMS EXISTING IN THE VERTICAL DISCRETIZATION OF THE HYDROSTATIC EQUATION AND IMPROVEMENT TESTS

Through analysis and numerical computation of ECMWF's discrete scheme of hydrostatic equation(Baede et al.1979),it has been found that in the case of equal △σ there exist systematic errors in the scheme.The error EΦ caused by taking the arithmetic mean of the geopotential heights of two adjacent half σ-levels as the geopotential height of the corresponding integer σ-level,increases with height and has an unacceptable maximum in the vicinity of the top of the atmosphere;however,the errors caused by the temperature treatment are generally small.On the other hand,if an uneven △σ-scheme in which the levels in the upper and lower atmosphere are denser than those in the middle atmosphere,is adopted,then EΦ can be much reduced.However,if the resolution of the original equal Art-scheme doubles,then EΦ can only be found to be much reduced in the troposphere and that in the vicinity of the atmospheric top is still unacceptable.Two numerical schemes for improvement have been presented. Of them one is the same as the ECMWF's scheme,but with equal △lnσ,and the other is to integrate the equation by the use of Tschebyscheff polynomials Tn and to adopt the zeros of TN as the atmospheric levels where N is the total number of levels.The results show that with both schemes the computational errors can be much reduced,especially the latter,due to the fact that the errors of three statistical types are generally less than the root mean square error of the geopotential heights reported in TEMP.