The Evolution of Dynamical Cores for Global Atmospheric Models

Abstract

The evolution of global atmospheric model dynamical cores from the first developments in the early 1960s to present day is reviewed. Numerical methods for atmospheric models are not straightforward because of the so-called pole problem. The early approaches include methods based on composite meshes, on quasi-homogeneous grids such as spherical geodesic and cubed sphere, on reduced grids, and on a latitude-longitude grid with short time steps near the pole, none of which were entirely successful. This resulted in the dominance of the spectral transform method after it was introduced. Semi-Lagrangian semi-implicit methods were developed which yielded significant computational savings and became dominant in Numerical Weather Prediction. The need for improved physical properties in climate modeling led to developments in shape preserving and conservative methods. Today the numerical methods development community is extremely active with emphasis placed on methods with desirable physical properties, especially conservation and shape preservation, while retaining the accuracy and efficiency gained in the past. Much of the development is based on quasi-uniform grids. Although the need for better physical properties is emphasized in this paper, another driving force is the need to develop schemes which are capable of running efficiently on computers with thousands of processors and distributed memory.Test cases for dynamical core evaluation are also briefly reviewed. These range from well defined deterministic tests to longer term statistical tests with both idealized forcing and complete parameterization packages but simple geometries. Finally some aspects of coupling dynamical cores to parameterization suites are discussed.

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

“…The efficiency and robustness achieved with SISL is not easily replaced by alternative choices [110], although the better scalability that can be exploited on future HPC architectures, due to less reliance on communication exchanges beyond nearest neighborhoods will work in favor of techniques with compact stencils, e.g. finite volume (FV) and spectral element methods (SEM), coupled with EBTI-based approaches.…”

An improved version of SLICE for conservative monotonic remapping on a C‐grid

“…Semi-Lagrangian (SL) transport schemes are an integral part of many atmospheric models (Williamson, 2007). They are robust and stable even at large Courant number.…”

A conservative Fourier‐finite‐element method for solving partial differential equations on the whole sphere

“…In the context of climate studies, a crucial additional requirement is that it be conservative: the discretized system should enforce the exact conservation of a discrete approximation of the total mass, momentum and, if possible, energy and enstrophy. For an in-depth review of the evolution of dynamical cores, the reader is referred to Williamson (2007). Although a single optimal scheme has not emerged yet, most dynamical cores in use today use one of two methods: the finite-difference method (Arakawa, 1966;Sadourny, 1975;Arakawa and Lamb, 1981) or the spectral-transform method (Orszag, 1970;Swarztrauber, 1996), both using a structured latitude-longitude grid.…”

“…This so-called 'pole problem' can be overcome by applying latitude-dependent zonal Fourier filters near the Poles. Although widely adopted, this fix remains 'unsatisfying' (Williamson, 2007) and may introduce discretization errors (Purser, 1988). Furthermore, Fourier filtering incurs an asymptotic cost of O(N 2 log N).…”

“…These exact conservation properties come 'for free' provided only that one adheres to the Galerkin framework and that the integrals can be computed exactly. Much current research on numerical schemes for dynamical cores has abandoned the latitude-longitude grid and focuses on the use of quasi-uniform grids with less severe singularities (Williamson, 2007;Rancic et al, 2008). In the present work, we instead keep the familiar latitude-longitude grid and design a new numerical method that is more accurate than finite differences and more efficient than the spectral transform, borrowing from the two approaches to achieve comparable stability and conservation properties.…”