A time-split, forward-backward numerical model for solving a nonhydrostatic and compressible system of equations

In this paper, we present a numerical procedure for solving a 2-dimensional, compressible, and nonhydrostatic system of equations. A forward-backward integration scheme is applied to treat high-frequency and internal gravity waves explicitly. The numerical procedure is shown to be neutral in time as long as a Courant–Friedrichs–Lewy criterion is met. Compared to the leapfrog-scheme most models use, this method involves only two time steps, which requires less memory and is also free from unstable computational modes. Hence, a time-filter is not needed. Advection and diffusion terms are calculated with a time step longer than sound-wave related terms, so that extensive computer time can be saved. In addition, a new numerical procedure for the free-slip bottom boundary condition is developed to avoid using inaccurate one-sided finite difference of pressure in the surface horizontal momentum equation when the terrain effect is considered. We have demonstrated the accuracy and stability of this new model in both linear and nonlinear situations. In linear mountain wave simulations, the model results match the corresponding analytical solution very closely for all three cases presented in this paper. The analytical streamlines for uniform flow over a narrow mountain range were obtained through numerical integration of Queney’s mathematical solution. It was found that Queney’s original diagram is not very accurate. The diagram had to be redrawn before it was used to verify our model results. For nonlinear tests, we simulated the famous 1972 Boulder windstorm and a bubble convection in an isentropic enviroment. Although there are no analytical solutions for the two nonlinear tests, the model results are shown to be very robust in terms of spatial resolution, lateral boundary conditions, and the use of the time-split scheme.