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Accurate Numerical Differencing near a Polar Singularity of a Skipped Grid

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  • 1 Cooperative Institute for Meteorological Satellite Studies, University of Wisconsin, Madison, Wisconsin
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Abstract

For a model using spherical coordinates the presence of a polar singularity presents a number of numerical problems that deserve careful attention. Retaining all zonal Fourier components near a pole necessitates an unduly short timestep if advection or adjustment are treated explicitly. Even if both processes are treated implicitly the resolution of very small structures would require inordinately more computational effort to satisfy the implicit equations here than at other more typical portions of the grid. This difficulty is usually avoided by reducing the longitudinal angular resolution near each pole by Fourier filtering or else by deploying a “skipped grid” with progressively fewer active grid points per latitude circle at higher latitudes. While successfully easing the restriction on the timestep, these measures degrade the accuracy of the zonal gradients in the immediate vicinity of each pole unless special precautions are taken.

This paper examines the scale-dependence, averaged over orientational of truncation error associated with conventional zonal interpolation and differencing formulae applied to the skipped grid. A remedy is proposed for the serious errors encountered and is shown to result in substantial improvements in the attainable accuracy. The method involves the smooth interpolation of zonally high frequency data to the innermost circle of latitude from the surrounding circle of data where the longitudinal resolution is higher.

Abstract

For a model using spherical coordinates the presence of a polar singularity presents a number of numerical problems that deserve careful attention. Retaining all zonal Fourier components near a pole necessitates an unduly short timestep if advection or adjustment are treated explicitly. Even if both processes are treated implicitly the resolution of very small structures would require inordinately more computational effort to satisfy the implicit equations here than at other more typical portions of the grid. This difficulty is usually avoided by reducing the longitudinal angular resolution near each pole by Fourier filtering or else by deploying a “skipped grid” with progressively fewer active grid points per latitude circle at higher latitudes. While successfully easing the restriction on the timestep, these measures degrade the accuracy of the zonal gradients in the immediate vicinity of each pole unless special precautions are taken.

This paper examines the scale-dependence, averaged over orientational of truncation error associated with conventional zonal interpolation and differencing formulae applied to the skipped grid. A remedy is proposed for the serious errors encountered and is shown to result in substantial improvements in the attainable accuracy. The method involves the smooth interpolation of zonally high frequency data to the innermost circle of latitude from the surrounding circle of data where the longitudinal resolution is higher.

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