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FREDERICK G. SHUMAN

Abstract

A method is developed for the design of finite-difference smoothing and filtering operators which meet pre-determined specifications, and which are applicable to automatic computing machinery. The general technique is to build complicated operators from the simplest types. The necessity for smoothing predicted fields of stream functions before inverting the balance equation for heights of isobaric surfaces is brought out.

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Frederick G. Shuman
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FREDERICK G. SHUMAN

Abstract

Two methods of solving the balance equation are outlined. Both methods have been used successfully on a daily operational basis at the Joint Numerical Weather Prediction Unit for a period of more than a year. Solutions were on the operational grid of 30 × 34 points spaced at 381-km. intervals.

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FREDERICK G. SHUMAN

Abstract

The physical inconsistency of geostrophic flow and small surface pressure tendencies is discussed. The frequently disastrous consequences in conventional geostrophic barotropic predictions are numerically identified by comparisons with experimental “semi-geostrophic” barotropic predictions from which the inconsistency has been removed. Effects of the inconsistency of the geostrophic wind field with the equations of motion are also quantitatively isolated by comparisons of semi-geostrophic predictions with predictions made with wind fields which satisfy the balance equation. It is concluded that the principal fault of the conventional geostrophic approximation lies in the violation of the continuity equation. Its lack of the dynamic effects expressed in the equations of motion seems also significant, but is less important.

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Frederick G. Shuman

Abstract

Excessive errors are committed in grids, in which mesh-lengths in the longitudinal direction are preserved in polar regions, if the hydrodynamic equations written in spherical coordinates are directly transformed into finite differences. The errors arise from the curvilinearity of the coordinate system. One alternative is the abandonment of spherical coordinates, with the retention of the constant mesh-length grid, an alternative which is not economical. The errors can be reduced by at least two orders of magnitude by adoption of a grid regular in latitude and longitude angle, with the consequent great space resolution in polar regions. Even the latter alternative may result in unacceptable truncation errors in finite-difference equations in spherical co-ordinates. It turns out, however, that if one departs from spherical coordinates to the extent of expressing velocity components in Cartesian coordinates on locally tangent planes, a further reduction of error by an order of magnitude is achieved, a reduction to perhaps acceptable levels.

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Frederick G. Shuman

Abstract

The first modern numerical weather prediction (NWP) models were developed for the computer that was announced in 1932 at the Institute for Advanced Study, in Princeton, New Jersey. Within 3 yr three agencies of the United States Government jointly created a numerical weather prediction service, but it was quickly discovered that current models had very serious defects. After considerable research, the first operationally effective model was achieved in 1958—a barotropic model covering most of the Northern Hemisphere. Over the years, models have evolved through multilevel filtered equation models and several primitive equation models. Analysis and data assimilation systems necessary for timeliness were also developed, and have likewise evolved. The result has been a revolution in weather forecasting.

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FREDERICK G. SHUMAN and LLOYD W. VANDERMAN

Abstract

A primitive-equation free-surface barotropic model was designed for the tropical belt. By the use of Shuman's difference system, experiments were made to test the effect of both approximate and correct boundary conditions on the forecast fields. Results are shown in the figures. With the correct boundary conditions a successful forecast was calculated without smoothing to 100 days.

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FREDERICK G. SHUMAN and JOHN D. STACKPOLE

Abstract

Numerical experimentation with various finite difference formulations of a particular set of differential equations incorporating a map scale factor indicates that the stability of the calculations is as dependent upon the manner in which the map factor is introduced as the form in which the dynamic terms of the equations are written.

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Frederick G. Shuman and John B. Hovermale

Abstract

In mid-1966 a new baroclinic numerical weather prediction model became operational at the National Meteorological Center, an event made possible by advances in computer and communications technology. The new model integrates directly the primitive (hydrostatic) hydrodynamic and thermodynamic equations, a departure from previous operational models whose central dynamic equation was that of conservation of vorticity. In its first fourteen months of operational use, it has resulted in highly significant improvements in the Center's products.

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ANDRÉ J. ROBERT, FREDERICK G. SHUMAN, and JOSEPH P. GERRITY JR.

Abstract

A rather general theory of nonlinear computational stability is reported. Instability is caused by both spatial and temporal high frequencies that, if not present initially, will appear from nonlinear interactions. It appears that through simple remedies relative stability, if not perfect stability, can be achieved.

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