Stable centered-difference schemes, based on an unstaggered a grid, that eliminate two-grid interval noise

The use of a centered-difference scheme on an unstaggered horizontal grid i n time-dependent atmospheric or oceanic models leads to spurious two-grid-i nterval wave solutions that may appear as small-scale noise and mask the ph ysically significant large-scale flow patterns. Although the problem is ind ependent of the time-difference schemes used in such models, the original t ime-difference schemes can be slightly modified to eliminate this problem. Following an established approach, the divergence held on the grid is modif ied in such a way that the height perturbation at a single grid point is pr opagated by gravity waves to all other grid points so that no spurious, two -grid-interval noise can develop. In this paper, a family of modified expli cit, two-step iterative, two-time-level schemes is developed and studied sy stematically for the unstaggered grid. The family of schemes is first developed for two-dimensional, nondispersive , surface gravity waves in a linearized shallow water model. The stability and phase properties of the original and the modified schemes are compared. The modified family of schemes is then implemented in a nonlinear shallow water model and its effectiveness is demonstrated by applying the model to an idealized forced initial value problem. Then the schemes are implemented in a multilevel, limited-area gridpoint NWP model. Results of 48-h time in tegrations using the model, once again, demonstrate the effectiveness of th e modified family of schemes.