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

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Hsu, Wu-Ron ; Sun, Wen-Yih (2011)

In this paper, we present a numerical procedure for solving a 2-dimensional, compressible, andnonhydrostatic system of equations. A forward-backward integration scheme is applied to treathigh-frequency and internal gravity waves explicitly. The numerical procedure is shown to beneutral 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 lessmemory 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 relatedterms, so that extensive computer time can be saved. In addition, a new numerical procedurefor the free-slip bottom boundary condition is developed to avoid using inaccurate one-sidedfinite difference of pressure in the surface horizontal momentum equation when the terraineffect is considered. We have demonstrated the accuracy and stability of this new model in bothlinear and nonlinear situations. In linear mountain wave simulations, the model results matchthe 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 obtainedthrough numerical integration of Queney’s mathematical solution. It was found that Queney’soriginal diagram is not very accurate. The diagram had to be redrawn before it was used toverify our model results. For nonlinear tests, we simulated the famous 1972 Boulder windstormand a bubble convection in an isentropic enviroment. Although there are no analytical solutionsfor the two nonlinear tests, the model results are shown to be very robust in terms of spatialresolution, lateral boundary conditions, and the use of the time-split scheme.DOI: 10.1034/j.1600-0870.2001.01178.x
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