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Stable Centered-Difference Schemes, Based on an Unstaggered A Grid, That Eliminate Two-Grid Interval Noise

Sajal K. KarBureau of Meteorology Research Centre, Melbourne, Australia

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Abstract

The use of a centered-difference scheme on an unstaggered horizontal grid in time-dependent atmospheric or oceanic models leads to spurious two-grid-interval wave solutions that may appear as small-scale noise and mask the physically significant large-scale flow patterns. Although the problem is independent of the time-difference schemes used in such models, the original time-difference schemes can be slightly modified to eliminate this problem. Following an established approach, the divergence field on the grid is modified in such a way that the height perturbation at a single grid point is propagated 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 explicit, two-step iterative, two-time-level schemes is developed and studied systematically 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 integrations using the model, once again, demonstrate the effectiveness of the modified family of schemes.

Corresponding author address: Dr. Sajal K. Kar, Bureau of Meteorology Research Centre, GPO Box 1289K, Melbourne 3001, Victoria, Australia.

Email: skar@BoM.gov.au

Abstract

The use of a centered-difference scheme on an unstaggered horizontal grid in time-dependent atmospheric or oceanic models leads to spurious two-grid-interval wave solutions that may appear as small-scale noise and mask the physically significant large-scale flow patterns. Although the problem is independent of the time-difference schemes used in such models, the original time-difference schemes can be slightly modified to eliminate this problem. Following an established approach, the divergence field on the grid is modified in such a way that the height perturbation at a single grid point is propagated 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 explicit, two-step iterative, two-time-level schemes is developed and studied systematically 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 integrations using the model, once again, demonstrate the effectiveness of the modified family of schemes.

Corresponding author address: Dr. Sajal K. Kar, Bureau of Meteorology Research Centre, GPO Box 1289K, Melbourne 3001, Victoria, Australia.

Email: skar@BoM.gov.au

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