On Certain Integral Constraints for the Time-Integration of Baroclinic Models

Abstract

On the basis of the general property of the vorticity equation that the net-production of vorticity vanishes in the average over a closed region it is in section 3 shown that provided this property should be fulfilled one should at the same time include or exclude the vertical advection of vorticity and the twisting effect in two-parameter models used for numerical weather prediction. The advection with the divergent part of the horizontal wind and the part of the divergence term which depends upon the relative vorticity has a similar opposing effect in the mean production of vorticity for the non-divergent level, but a vanishing production of thermal vorticity can only be obtained if one either include the advection with the divergent wind and the complete divergence term or exclude the advection with divergent wind, the part of the divergence term depending upon the relative vorticity and consider the Coriolis parameter in the remaining part of the divergence term as a constant. In section 4 it is shovn that the mean value over a closed region of the relative topography remains constant in a two-parameter model provided the vertical stability parameter is constant. It is further shown that the thermodynamic equation in the adiabatic case is far less sensitive to inconsistencies than the vorticity equation. Section 5 contains a comparison between the growth of “rotational energy” in general and simplified two-parameter models. It is shown that already the most simple model contains approximately the correct growth rate. A similar comparison with respect to the rate of “occlusion” is made in section 6 with a similar result. Section 7 contains a comparison between the theoretical results and the experience from practical computations. The mechanism for production of kinetic energy of the nondivergent part of the wind is discussed in section 8. DOI: 10.1111/j.2153-3490.1959.tb00004.x