Comparison of Space and Time Errors in Spectral Numerical Solutions of the Global Shallow-Water Equations

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  • 1 Bureau of Meteorology Research Centre, Melbourne, Australia
  • | 2 Cooperative institute for Research in the Atmosphere, Fort Collins, Colorado NOAA/ERL Forecast Systems Laboratory, Boulder, Colorado
  • | 3 Bureau of Meteorology Research Centre, Melbourne, Australia
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Abstract

The convergence of spectral model numerical solutions of the global shallow-water equations is examined as a function of the time step and the spectral truncation. The contributions to the errors due to the spatial and temporal discretizations are separately identified and compared. Numerical convergence experiments are performed with the inviscid equations from smooth (Rossby-Haurwitz wave) and observed (R45 atmospheric analysis) initial conditions, and also with the diffusive shallow-water equations. Results are compared with the forced inviscid shallow-water equations case studied by Browning et at. Reduction of the time discretization error by the removal of fast waves from the solution using initialization is shown. The effects of forcing and diffusion on the convergence are discussed. Time truncation errors are found to dominate when a feature is large scale and well resolved; spatial truncation errors dominate-for small-scale features and also for large scales after the small scales have affected them. Possible implications of these results for global atmospheric modeling are discussed.

Abstract

The convergence of spectral model numerical solutions of the global shallow-water equations is examined as a function of the time step and the spectral truncation. The contributions to the errors due to the spatial and temporal discretizations are separately identified and compared. Numerical convergence experiments are performed with the inviscid equations from smooth (Rossby-Haurwitz wave) and observed (R45 atmospheric analysis) initial conditions, and also with the diffusive shallow-water equations. Results are compared with the forced inviscid shallow-water equations case studied by Browning et at. Reduction of the time discretization error by the removal of fast waves from the solution using initialization is shown. The effects of forcing and diffusion on the convergence are discussed. Time truncation errors are found to dominate when a feature is large scale and well resolved; spatial truncation errors dominate-for small-scale features and also for large scales after the small scales have affected them. Possible implications of these results for global atmospheric modeling are discussed.

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