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Andrew N. Staniforth
and
Herschel L. Mitchell

Abstract

A barotropic primitive-equation model using the finite-element method of space discretization is generalized to allow variable resolution. The overhead incurred in going from a uniform mesh to a variable mesh having the same number of degrees of freedom is found to be approximately 20% overall.

The variable-mesh model is used with several grid configurations, each having uniform high resolution over a specified area of interest and lower resolution elsewhere to produce short-term forecasts over this area without the necessity of high resolution everywhere. It is found that the forecast produced on a uniform high-resolution mesh can be essentially reproduced for a limited time over the limited area by a variable-mesh model having only a fraction of the number of degrees of freedom and requiring significantly less computer time. As expected, the period of validity of forecasts on variable meshes can be lengthened by refining the mesh in the outer region.

It is concluded that from the point of view of efficiency, accuracy and stability the variable-mesh finite-element technique appears to be well-suited to the practical problem of limited area/time forecasting.

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Andrew N. Staniforth
and
Roger W. Daley

Abstract

A baroclinic primitive equations model is formulated using a variable resolution finite-element discretization in all three space dimensions. The horizontal domain over which the model is integrated is a rectangle on a polar stereographic projection which approximately covers the Northern Hemisphere. A wall boundary condition is imposed at this rectangular boundary giving rise to a well-posed initial boundary value problem. The mesh is specified to be of Cartesian product form with arbitrary non-uniform spacing. By choosing the mesh to be uniformly high over an area of interest and degrading smoothly away from this area, it is possible to use the model to produce a high-resolution local forecast for a limited time period. This choice of mesh avoids the noise problems of a so-called nested grid. A semi-implicit time discretization is used for efficiency. Some results for forecast periods of 24 and 48 h are also given to demonstrate its viability in an operational context.

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Andrew N. Staniforth
and
Herschel L. Mitchell

Abstract

A barotropic primitive equation model using the finite-element method of space discretization is presented. The semi-implicit method of time discretization is implemented by taking a suitable form of the governing equations. Finite-element forecasts are then compared with those from both finite-difference and spectral models. From the point of view of both efficiency and accuracy it is concluded that the finite-element method of space discretization appears to he viable for the practical problem of numerical weather prediction.

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Andrew N. Staniforth
and
Roger W. Daley

Abstract

A finite-element formulation for the vertical structure of primitive equation models has been developed. The finite-element method is a variant of the Galerkin procedure in which the dependent variables are expanded in a finite, set of basis functions and then the truncation error is orthogonalized to each of the basis functions. In the present case, the basis functions are Châpeau functions in sigma, the vertical coordinate. The procedure has been designed for use with a semi-implicit time discretization algorithm.

Although this vertical representation has been developed for ultimate implementation in a three-dimensional finite-element model, it has been first tested in a spherical harmonic, baroclinic, primitive equations model. Short-range forecasts made with this model are very encouraging.

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