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On the Accuracy of Omega Diagnostic Computations

Stanley L. BarnesEnvironmental Research Laboratories, National Oceanic and Atmospheric Administration, Boulder, CO 80303

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Abstract

The accuracies of algorithms used to compute quasi-geostrophic diagnostic parameters such as Hoskins' Q-vector and its divergence are evaluated. Analytically-determined height values are invoked at grid points representing three pressure surfaces, and finite difference approximations to third and lower order derivatives are compared with analytic values. Errors from these approximations are found to be virtually identical to those predicted by a mathematical analysis of the centered difference scheme. The magnitudes of finite difference errors are a function of wavelength, being acceptably small for waves sampled nine or more times per wavelength. Interpolation of grid point values from analytically determined observations at a typical array of rawinsonde stations produces diagnostic results that, while they contain many distortions owing mainly to data sparseness still contain significant portions of the signal, as determined by a two-dimensional spectral estimation technique using Fourier analysis. With reference to results from a previous case study, it is concluded that meaningful meteorological information was diagnosed at wavelengths of about 1250 km and larger. However, the results also suggest that it may not be appropriate to compute vertical motions per se for the smaller wavelengths because of interpolation-induced uncertainties in the diagnosed fields of geostrophic forcing. Appendices contain derivations of theoretical errors in second-order, centered difference estimations of first, second and third derivatives, and explain the procedures for obtaining spectral estimates in limited domains for two-dimensional fields.

Abstract

The accuracies of algorithms used to compute quasi-geostrophic diagnostic parameters such as Hoskins' Q-vector and its divergence are evaluated. Analytically-determined height values are invoked at grid points representing three pressure surfaces, and finite difference approximations to third and lower order derivatives are compared with analytic values. Errors from these approximations are found to be virtually identical to those predicted by a mathematical analysis of the centered difference scheme. The magnitudes of finite difference errors are a function of wavelength, being acceptably small for waves sampled nine or more times per wavelength. Interpolation of grid point values from analytically determined observations at a typical array of rawinsonde stations produces diagnostic results that, while they contain many distortions owing mainly to data sparseness still contain significant portions of the signal, as determined by a two-dimensional spectral estimation technique using Fourier analysis. With reference to results from a previous case study, it is concluded that meaningful meteorological information was diagnosed at wavelengths of about 1250 km and larger. However, the results also suggest that it may not be appropriate to compute vertical motions per se for the smaller wavelengths because of interpolation-induced uncertainties in the diagnosed fields of geostrophic forcing. Appendices contain derivations of theoretical errors in second-order, centered difference estimations of first, second and third derivatives, and explain the procedures for obtaining spectral estimates in limited domains for two-dimensional fields.

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