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  • Author or Editor: H. A. Riphagen x
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I. M. Navon
and
H. A. Riphagen

Abstract

An alternating-direction implicit finite-difference scheme is developed for solving the nonlinear shallow-water equations in conservation-law form.

The algorithm is second-order time accurate, while fourth-order compact differencing is implemented in a spatially factored form. The application of the higher order compact Padé differencing scheme requires only the solution of either block-tridiagonal or cyclic block-tridiagonal coefficient matrices, and thus permits the use of economical block-tridiagonal algorithms. The integral invariants of the shallow-water equations, i.e., mass, total energy and enstrophy, are well conserved during the numerical integration, ensuring that a realistic nonlinear structure is obtained.

Largely in an experimental way, two methods are investigated for determining stable approximations for the extraneous boundary conditions required by the fourth-order method. In both methods, third-order uncentered differences at the boundaries are utilized, and both preserve the overall fourth-order convergence rate of the more accurate interior approximation.

A fourth-order dissipative term was added to the equations to overcome the increased aliasing due to the fourth-order method. Alternatively, Wallington and Shapiro low-pass filters were applied.

The numerical integration of the shallow-water equations is performed in a channel corresponding to a middle-latitude band. A linearized version of this method is shown to be unconditionally stable.

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H. A. Riphagen
,
C. L. Bruyère
,
W. Jordaan
,
E. R. Poolman
, and
J. D. Gertenbach

Abstract

This paper describes experiments with the National Centers for Environmental Prediction (NCEP) regional Eta Model, which runs operationally at the South African Weather Bureau. Experiments were designed to assess manual methods of improving the model representation of the surface topography at low resolutions where this might be feasible. The manual terrain modifications, applied in the then-operational 80-km Eta Model, involved either indirect alteration of the terrain through judicious rearrangement of the model coordinate surfaces, or the direct modification of the model terrain to accommodate meteorologically important features of the actual topography not well represented in the model. Also studied was the effect on the model terrain of changes in the horizontal grid configuration, with a potential for exploitation at higher model resolutions. Trials at higher resolutions also became available as part of a stepwise implementation of an upgrade to the 80-km system. This upgrade contained the important model enhancements installed at NCEP in January 1996 (new soil hydrology, land surface and viscous sublayer parameterization, and revised turbulence scheme) and February 1997 (notably changes to the radiation package and land surface model). A case of light nocturnal rainfall over Gauteng, linked to a topographical trigger and poorly forecast on the day, was the focus of a study to demonstrate the viability of the methods. Specifically, the link between changes in model topography and in precipitation forecasts from the model was investigated.

As expected, the model enhancement contained in the upgrade package provided the greatest impact on the precipitation predictions, and higher horizontal resolutions generally led to better representations of the terrain and more realistic forecasts. Nevertheless, the possibility exists of improving the high-resolution model topography yet further through modifications to the grid structure. If, however, higher horizontal resolutions are not affordable, it seems from this prototype study that slight grid shifts or manual modification of the model terrain could be useful and viable options. There is some potential in the judicious choice of the vertical coordinate structure, although what is good for one region might prove bad for another. There seems to be more potential in the manual correction of meteorologically significant deficiencies in the model terrain. These approaches are likely to be beneficial also in the presence of higher vertical resolution, which in itself may not be sufficient to improve the topography. Altering the terrain by hand, even at low resolutions, is a laborious task and only feasible at low resolutions but with the Eta Model being run on smaller machines, in centers with limited resources, this option seems to deserve consideration.

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