A Variable Resolution Global Model Based upon Fourier and Finite Element Representation

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Abstract
We describe the development and preliminary testing of a numerical scheme designed to predict the global circulation which can also telescope into local subdomains of enhanced vertical and horizontal resolution. The accuracy of the method appears intermediate to the accuracy of purely spectral and grid point models, but it is especially well suited to study certain practical predictability problems within limited area domains. The approach is based upon separate Galerkin approximations in longitude and latitude. The longitude variation is discretized in terms of truncated Fourier series, while the latitude structure of the Fourier amplitudes is depicted as sums of piecewise continuous linear functions (finite elements). The vertical structure is also described in terms of finite elements. The technique is especially well suited to fine resolution of polar caps within which a given wavenumber truncation ensures enhanced local resolution in longitude, and where a refined element size can also be implemented to improve latitude resolution. These polar lenses can be rotated over any geographical region of special interest and therefore serve to enhance local resolution in subdomains that have completely general two-way interactions with global scales. The numerical difficulties generally associated with polar singularities do not pose special problems in the present approach. This is illustrated with a series of Rossby-Haurwitz wave examples Cases with divergent flow and multilevel applications each require certain additional modifications of the horizontal treatment. The multilevel version has a fully interactive atmosphere, interface, and subsurface. The vertical fluxes throughout the atmospheric portion of the model are formulated in terms of a turbulence closure which explicitly predicts evolution of the troposphere, the planetary boundary layer, and the surface boundary layer. Surface turbulent fluxes are computed as the flux that naturally appears at the lowest level of the model, without separate parameterization in terms of bulk transfer coefficients. The sensitivity of these calculations to vertical resolution and to the frequency of updating the infrared flux is described. It appears that as few as three appropriately positioned forecast levels in the surface and planetary boundary layers may be sufficient to many applications. The principal deficiencies of the approach are the relative complexity introduced by the mixed numerical treatment and a rather high computational overhead. However, the method is especially well suited to address questions of regional predictability in which limited area models have produced rather perplexing results that may be partly attributable to artificial lateral boundaries. A preliminary study of this is made for the case of topographically bound low-level jets.

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