Forward-in-time differencing for fluids: Nonhydrostatic modeling of fluid motions on a sphere

Traditionally, numerical models for simulating planetary scale weather and climate employ the hydrostatic primitive equations--an abbreviated form of Navier-Stokes` equations that neglect vertical accelerations and use simplified Coriolis forces. Although there is no evidence so far that including nonhydrostatic effects in global models has any physical significance for large scale solutions, there is an emerging trend in the community toward restoring Navier-Stokes` equations (or at least their less constrained forms) in global models of atmospheres and oceans. The primary motivation is that state-of-the-art computers already admit resolutions where local nonhydrostatic effects become noticeable. much of this present research aims to improve the design of a high-performance numerical model for simulating the flows of moist (and precipitating), rotating, stratified fluids past a specified time-dependent irregular lower boundary. This model is representative of a class of nonhydrostatic atmospheric codes that employs the anelastic equations of motion in a terrain-following curvilinear framework, and contains parallel implementations of semi-Lagrangian and Eulerian approximations selectable by the user. The model has been employed in a variety of application; the quality of results suggest that modern nonoscillatory forward-in-time (NFT) methods are superior to the more traditional centered-in-time-and-space schemes, in terms of accuracy, computational efficiency, flexibility and robustness. The authors have extended the Cartesian NFT model to a mountainous sphere and, consequently, have dispensed with the traditional geophysical simplifications of hydrostaticity, gentle terrain slopes, and weak rotation. In this paper, they discuss the algorithmic design, relative efficiency and accuracy of several different variants (hydrostatic, nonhydrostatic, implicit, explicit, etc.) of the NFT global model. They substantiate their theoretical discussions with the results of simulations of idealized global orographic flows and climates