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Comments on “Truncation Errors in Finite-Difference Estimates of Geostrophic Wind and Relative Vorticity”

Annette M. LarioDepartment of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104

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Donald J. PerkeyDepartment of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104

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Shing YohDepartment of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104

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Jing GuoDepartment of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104

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Abstract

Peppler and Smith (1984) discussed truncation errors associated with second-order and fourth-order finite difference approximations used to calculate the geostrophic wind and relative vorticity. They found that these errors were, in general, smaller for longer wavelengths, finer-grid resolution, and fourth-order differencing. Second-order differencing produced fields that were numerically less than their corresponding analytic values and yielded errors which decreased with reduced grid interval. Fourth-order differencing decreased the errors when the grid interval was reduced, but only while the wavelength was ten limes or more greater than the grid interval.

Results presented here indicate that the wind speed and Vorticity errors estimated by the fourth-order scheme decreased when the grid interval was decreased independent of wavelength. Comparison with Peppler and Smith's actual computations showed only one difference: in their finite-(difference equations the coefficients were truncated to the nearest thousandth (for example, 0.083 was used in place of 1/12). On the other hand, we used 1.0/12.0 and allowed the computer's word length to determine the value of the coefficient, thus, preserving greater accuracy.

Abstract

Peppler and Smith (1984) discussed truncation errors associated with second-order and fourth-order finite difference approximations used to calculate the geostrophic wind and relative vorticity. They found that these errors were, in general, smaller for longer wavelengths, finer-grid resolution, and fourth-order differencing. Second-order differencing produced fields that were numerically less than their corresponding analytic values and yielded errors which decreased with reduced grid interval. Fourth-order differencing decreased the errors when the grid interval was reduced, but only while the wavelength was ten limes or more greater than the grid interval.

Results presented here indicate that the wind speed and Vorticity errors estimated by the fourth-order scheme decreased when the grid interval was decreased independent of wavelength. Comparison with Peppler and Smith's actual computations showed only one difference: in their finite-(difference equations the coefficients were truncated to the nearest thousandth (for example, 0.083 was used in place of 1/12). On the other hand, we used 1.0/12.0 and allowed the computer's word length to determine the value of the coefficient, thus, preserving greater accuracy.

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