Conservative Scheme for a Compressible Nonhydrostatic Model with Moist Processes

Abstract

A conservative scheme for a compressible nonhydrostatic model including moist processes is formulated and is tested for experiments involving a squall line. The scheme is based on the flux form equations of total density, momentum, total energy, and the densities of water substance. The time-splitting scheme is used for the temporal scheme. In the small time step integration, the horizontal components of momentum are explicitly integrated, while the vertical components of momentum, density, and total energy are implicitly integrated. In particular, the flux form equation for the total energy is used to guarantee the conservation of the total energy. The internal energy is obtained by subtracting the kinetic energy and the potential energy from the total energy. This method is advantageous for the energy budget analysis. Only the warm rain cloud process is included for cloud physics. Using the squall-line experiments, the water budget and the energy budget are diagnosed and it is confirmed that the conservation of water and total energy is well satisfied. As a quantitative improvement, more accurate formulas are used for the thermodynamics of the moist atmosphere by taking account of the effects of specific heats of the water substance and the temperature dependency of latent heat. These effects are generally neglected in most numerical models. If accurate moist thermodynamics are used, the total rain is reduced by more than 10% in comparison to the case when the simplified thermodynamics are used. The transportations of physical quantities due to rain are also appropriately introduced using a higher-order advection scheme. In the flux-form formulation, it is found that the change in energy due to the transportation of rain cannot be neglected in general, while that in momentum could be negligible.