# Current Development in Shallow Water Models on the Sphere

## Problem Description

Modern weather prediction and climate simulation, the accuracy and timeliness of which are important to both public safety and world economy, are frequently based on global circulation models (GCMs). A GCM consists of three interacting components: data-assimilation, which incorporates data from observations into the model; numerical dynamics, which is concerned with the numerical solution of the so-called primitive equations of the hydrodynamics in the atmosphere; and physical parametrization, which incorporates into the model physical processes not represented in the primitive equations, for instance, radiation, cumulus convection, large-scale precipitation and turbulence. This workshop will focus on the numerical dynamics component. Until the mid 1990's, GCMs were typically discretized on regular latitude-longitude grids. On such grids, the meridians converge toward the poles and the physical distance between grid lines becomes small while the resolution in the tropical region remains coarse. When an Eulerian integration method is used, the atypically high resolution near the poles imposes a prohibitive restriction on the time-step size, thus giving rise to one of the ``pole problems.'' The resulting scalability problem has become more pressing with the demand for increasing grid resolutions initiated by improved hardware and physical parametrization schemes. Therefore, many groups have developed non-regular grid types and new numerical techniques aiming at a (quasi)-uniform cell distribution (see, e.g. [2,8]). Others focused on filtering techniques for suppressing irrelevant high-frequency waves [7] or on efficient implicit time integration methods [1,5,6]. Recently, many new numerical methods have been introduced in operational GCMs. With respect to the spatial discretization schemes, there are the Integrated Forecast System (IFS)-model of the European Centre for Mediuma-Range Weather Forecasts (ECMWF) and the Community Climate Model (CCM)-model operational at the National Center for Atmospheric Research (NCAR), which incorporate a spectral transform method; the Global Environmental Multiscale (GEM)-model of the Canadian Meteorological Centre, which applies a variable-resolution cell-integrated finite element scheme; and the GME model of the German Weather Service (DWD) [4] and the HIRLAM model developed by a consortium of several European weather services [3], which adopt central finite-difference schemes. All the modification improved the models, but a lot of unresolved problems remain. For instance, non-regular grid types can introduce an additional algebraic equation for horizontal motion. Furthermore, implicit time integration methods and filtering techniques can suppress relevent high frequency waves. This workshop aims at the indentification, understanding and possible solution of the typical problems related to these different approaches. The workshop schedule is designed to support this objective. In 1992, Williamson et al. [9] proposed a testset, based on the shallow water equations (SWEs), for testing new numerical methods intended for the solution of GCMs. The SWEs, which describe the inviscid flow of a thin layer of fluid in two dimensions and serve as a first prototype of the partial differential equations describing the horizontal dynamics of the atmosphere, have been used for many years by the atmospheric modeling community as a vehicle for testing promising numerical methods for solving atmospheric and oceanic problems. The simplicity of the SWEs facilitates a thorough investigation of the numerical method and their complexity, which captures important characteristics present in more comprehensive atmospheric models, allows the interactions between different physical quantities to be analyzed. In this workshop, comparisons between methods will be based on the SWEs testset.## Bibliography

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- Källén, E., 1996: HIRLAM documentation manual. Available from Erland Källén, SMHI, S-60176 Norrköping, Sweden
- Majewski, D., 1996: Documentation of the New Global Model (GME) of the DWD, Programmdokumentation, DWD-Abteilung Forschung und Entwicklung, Offenbach.
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- Taylor, M., J. Tribbia, and M. Iskandarani, 1997: The Spectral Element Method for the Shallow Water Equations on the Sphere, J. Comput. Phys., 130, 92-108.
- Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob and P. N. Swarztrauber, 1992: A Standard Test Set for Numerical Approximations to the Shallow Water Equations in Spherical Geometry, J. Comput. Phys., 102, 211-224