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W. Lawrence Gates

Abstract

After further experiments with a pilot case reported earlier, the results are summarized for a selected ten-day series of predictions obtained over the northern hemisphere by conventional numerical methods with the barotropic model. The overall accuracy of the forecasts is found to decay steadily with increasing forecast period up to 72 hours. In comparison with a series of barotropic forecasts prepared earlier for North America only, the hemispheric predictions are shown to be free of serious boundary-condition error in the middle-latitude regions of major synoptic activity. Outstanding among the errors remaining in the hemispheric integrations, however, are those due to the variation of the density of observational data (especially serious over the Pacific and Asiatic regions), those caused by excessive anticyclogenesis, those due to truncation error and the lack of smoothing, and those inherent in the model's neglect of baroclinic development. Research is in progress on these and other errors, in an effort to improve further the resolution of numerical prediction methods.

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W. Lawrence Gates

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Various measures of static stability in the atmosphere are reviewed and their uses briefly discussed. The mean vertical distribution of nine stability measures is given for 100-mb tropospheric layers and for selected stratospheric layers. The average geographical distribution over the United States is also discussed and illustrated for the measure – Tθ−1δθ/δp. The seasonal differences in stability distribution are discussed from the January and July average data for forty-five United States radiosonde stations.

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W. Lawrence Gates

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W. Lawrence Gates

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W. Lawrence Gates

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W. Lawrence Gates

Abstract

The truncation error, stability and convergence properties of various finite-difference formulations of the one-dimensional barotropic vorticity equation are considered, and analytic solutions of the difference equations for simple harmonic initial conditions are presented. With conventional centered space differences, the schemes considered may be classified according to the method of time differencing as the forward difference case (unstable), the first-forward-then-centered difference case (conditionally stable), and the implicit difference case (unconditionally stable). The first-forward-then-centered difference scheme, corresponding to that commonly employed in meteorological numerical integration, gives rise to an oscillation phenomenon in both the amplitude and phase speed of the solution, which is most serious for a small space mesh, a large time mesh, and for the shorter wavelength disturbances. In each difference scheme considered, the truncation error leads to a cumulative phase departure of the difference solution relative to the true solution, an effect which is approximately proportional to the cube of the wavelength.

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W. Lawrence Gates

The Atmospheric Model Intercomparison Project (AMIP) is an international effort to determine the systematic climate errors of atmospheric models under realistic conditions, and calls for the simulation of the climate of the decade 1979–1988 using the observed monthly averaged distributions of sea surface temperature and sea ice as boundary conditions. Organized by the Working Group on Numerical Experimentation as a contribution to the World Climate Research Programme, AMIP involves the international atmospheric modeling community in a major test and intercomparison of model performance; in addition to an agreed-to set of monthly averaged output variables, each of the participating models will generate a daily history of state. These data will be stored and made available in standard format by the Program for Climate Model Diagnosis and Intercomparison at the Lawrence Livermore National Laboratory. Following completion of the computational phase of AMIP in 1993, emphasis will shift to a series of diagnostic subprojects, now being planned, for the detailed examination of model performance and the simulation of specific physical processes and phenomena. AMIP offers an unprecedented opportunity for the comprehensive evaluation and validation of current atmospheric models, and is expected to provide valuable information for model improvement.

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W. LAWRENCE GATES

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By the use of two separate mesh sizes instead of one in computing finite differences, an extrapolation to effective “zero” mesh size may be made in order to reduce the truncation error, as originally suggested by Richardson. This technique of “difference extrapolation” is applied to the estimation of individual differentials in the barotropic vorticity equation, and here corresponds to the use of “second-order” finite differences. The truncation-induced phase speed lag of the difference solution relative to the true solution is shown to be systematically reduced, especially for the shorter waves. Next, the extrapolation is applied with two separate solutions of the barotropic difference equation, with the result that the phase speeds are further improved, but at the expense of an amplitude distortion of about 10 percent. This amplitude distortion may be removed for a particular wavelength, and a small further phase speed improvement obtained, but the amplitude distortion remains for other wavelengths. These methods of “solution extrapolation” are therefore felt to be unsuitable for routine use. The method of “difference extrapolation,” however, preserves the solution's amplitude, and if used in conjunction with a suitable smoothing procedure should result in a net error reduction for those waves resolved by the mesh and retained by the smoothing.

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W. Lawrence Gates

Abstract

By juxtaposition of a series of one-dimensional solutions at adjacent latitudes, numerical forecasts may easily and inexpensively be incorporated into forecasting routine. Forecasts for 140 degrees of longitude, for the region 30 to 70°N, can be prepared in 60 to 90 minutes by one person. It is suggested that, for some purposes of forecasting and synoptic research, these calculations may be used to approximate the solutions obtained with the aid of electronic computing equipment, with a considerable saving of time and expense. In application to the equivalent-barotropic model of Charney and Eliassen, the accuracy of the method is comparable to that of synoptic forecasts prepared in the conventional manner. The latter are presented for comparison, in a set of four 24-hour forecasts.

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W. Lawrence Gates

Abstract

The primitive hydrostatic equations for a rectangular homogeneous ocean with a free surface on a β-plane are integrated numerically for 60 days from an initial state of rest and undisturbed depth of 400 m. A zonal wind stress (maximum 2 dyn cm−2) and a lateral eddy viscosity (108 cm2 sec−1) are assumed. A series of transient Rossby waves of approximately 1000-2000 km in length form in the central and eastern basin, and undergo a well-marked life cycle of amplification and decay as they propagate westward at ∼1 m sec−1 relative to the zonal current. The northward boundary current in the west (∼1 m sec−1) and the counter-currents in the northwest (∼10 cm sec−1) may be identified as the first stationary members of a continuing series of waves, with subsequent transients showing characteristics of reflected Rossby waves and reaching progressively smaller maximum amplitudes. The standing wave pattern (wavelength ∼600 km) in the north-west is a characteristic nonlinear effect, and is associated with the meridional tilt displayed by the transients and the resultant (nonlinear) poleward eddy transport of zonal momentum. Near geostrophic equilibrium is maintained throughout, with the meridional Ekman flow of the order of a few centimeters per second. After a spin-up period of about 12 days, the surface potential and total kinetic energy display damped oscillations with the free period of approximately 16 days, with (long) surface gravity waves not significantly present.

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