A one-dimensional prognostic cloud model has been developed for possible use in a Cumulus Parameterization Scheme (CPS). In this model, the nonhydrostatic pressure, entrainment, cloud microphysics, lateral eddy mixing and vertical eddy mixing are included, and their effects are discussed. The inclusion of the nonhydrostatic pressure can (1) weaken vertical velocities, (2) help the cloud develop sooner, (3) help maintain a longer mature stage, (4) produce a stronger overshooting cooling, and (5) approximately double the precipitation amount. The pressure perturbation consists of buoyancy pressure and dynamic pressure, and the simulation results show that both of them are important. We have compared our simulation results with those from Ogura and Takahashi's one-dimensional cloud model, and those from the three-dimensional Weather Research and Forecast (WRF) model. Our model, including detailed cloud microphysics, generates stronger maximum vertical velocity than Ogura and Takahashi's results. Furthermore, the results illustrate that this one-dimensional model is capable of reproducing the major features of a convective cloud that are produced by the three-dimensional model when there is no ambient wind shear.

A fully compressible, three-dimensional, nonhydrostatic model is developed using a semi-implicit scheme to avoid an extremely small time step. As a result of applying the implicit scheme to high-frequency waves, an elliptic partial differential equation (EPDE) has been introduced. A multigrid solver is applied to solve the EPDEs, which include cross-derivative terms due to terrain-following coordinate transformation. Several experiments have been performed to evaluate the model as well as the performance of the scheme with respect to tolerance number, relaxation choice, sweeps of prerelaxation and postrelaxation, and a flexible hybrid coordinate (FHC). An FHC with two functions (base and deviation functions) is introduced. The basic function provides constant vertical grid spacing required in the multigrid solver, while the deviation function helps to adjust the vertical resolution.