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John Thuburn

Abstract

A nondivergent barotropic mode1 and a shallow-water model are presented that exploit a high-order shape-preserving scheme for the advection of vorticity or potential vorticity as well as tracers. The dissipation associated with the advection scheme is found to be due to the spreading of features as they are advected across a finite-resolution grid. The strength and scale selectivity of this dissipation are quantified using some simple tests. The resolved tracer variance and (potential) enstrophy are not conserved by the advection scheme; this can be interpreted as a cascade to unresolved scales. In simulations of turbulent mixing, satisfactory cascade of tracer variance, energy, and (potential) enstrophy are obtained without the need for any special parameterization of the cascade to unresolved scales.

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John Thuburn

Abstract

A new global shallow-water model has been developed. It uses a hexagonal–icosahedral grid, potential vorticity as a prognostic variable, and a conservative, shape-preserving scheme for advection of mass, potential vorticity, and tracers. A semi-implicit time scheme is used so that the maximum time step for stable integrations is limited by the advection speed rather than the gravity wave phase speed. This combination of numerical methods avoids some of the major problems of more traditional numerical methods, such as pole problems, and spurious oscillations and negatives in advected quantities. Sample results from a standard set of test cases are presented to illustrate the model’s performance. In a pure advection test case the model’s advection scheme shows good isotropy and phase-speed properties, but it is a little diffusive. In the remaining test cases the model’s overall accuracy is comparable to that of other gridpoint models for which results are available. Two sources of error are noted. One is the dissipation inherent in the advection scheme, which is estimated to be significantly stronger than the dissipation usually imposed in climate models of comparable resolution. The other is the grid structure, which leads to conspicuous symmetry errors in test cases where the true solution is symmetrical. The symmetry errors appear to arise because the hexagonal grid boxes are not perfectly regular but are somewhat distorted, particularly in certain regions of the grid, leading to larger truncation errors in the advection scheme in those regions.

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John Thuburn

Abstract

Three approaches to building one-dimensional shape-preserving advection schemes, based on TVD (total variation diminishing) schemes, on positive schemes, and on the universal limiter, are shown to lead to the same constraints on the fluxes between grid boxes. Thus, although they have slightly different conceptual bases, the three approaches lead to mathematically equivalent schemes.

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John Thuburn
and
Andrew Staniforth

Abstract

Discretizations of the linearized shallow-water equations on a spherical C grid are considered. Constraints on the schemes' coefficients that ensure conservation of mass, angular momentum, and energy are derived. These results generalize previously published results to the case of nonuniform and rotated grids (but are restricted to the linearized equations). Grids with υ stored at the poles and grids with u and h stored at the poles are both considered. Energy conservation is shown to be problematic for grids with u and h at the poles.

It is also shown that an inappropriate averaging of the Coriolis terms leads to a misrepresentation of the Rossby modes with shortest meridional scale. The appropriate averaging is shown to be compatible with the constraints required for conservation, and, indeed, the energy-conserving averaging of the Coriolis terms improves the dispersion properties of Rossby modes.

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Hilary Weller
,
John Thuburn
, and
Colin J. Cotter

Abstract

Currently, most operational forecasting models use latitude–longitude grids, whose convergence of meridians toward the poles limits parallel scaling. Quasi-uniform grids might avoid this limitation. Thuburn et al. and Ringler et al. have developed a method for arbitrarily structured, orthogonal C grids called TRiSK, which has many of the desirable properties of the C grid on latitude–longitude grids but which works on a variety of quasi-uniform grids. Here, five quasi-uniform, orthogonal grids of the sphere are investigated using TRiSK to solve the shallow-water equations.

Some of the advantages and disadvantages of the hexagonal and triangular icosahedra, a “Voronoi-ized” cubed sphere, a Voronoi-ized skipped latitude–longitude grid, and a grid of kites in comparison to a full latitude–longitude grid are demonstrated. It is shown that the hexagonal icosahedron gives the most accurate results (for least computational cost). All of the grids suffer from spurious computational modes; this is especially true of the kite grid, despite it having exactly twice as many velocity degrees of freedom as height degrees of freedom. However, the computational modes are easiest to control on the hexagonal icosahedron since they consist of vorticity oscillations on the dual grid that can be controlled using a diffusive advection scheme for potential vorticity.

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Zhengxin Zhu
,
John Thuburn
,
Brian J. Hoskins
, and
Peter H. Haynes

Abstract

A vertical discretization of the primitive equations in a general vertical coordinate is described that enables a primitive equation model to use terrain-following sigma levels near the ground and isentropic levels higher up, with a smooth transition region in between. Therefore, it combines many of the advantages of the computational efficiency of σ coordinates and the predictive and diagnostic potential of θ coordinates, and should be particularly useful for general circulation models to be used for studies of stratosphere-troposphere exchange and middle-atmosphere transport of trace gases. It is shown that the semi-implicit time scheme can be used in a straightforward manner with this discretization. A discussion is given of how to optimize the transition from sigma levels to isentropic levels so as to avoid model levels crossing each other. A numerical problem caused when very shallow, very strong inversions occur in the temperature field is countered by a form of vertical-scale selective dissipation. Baroclinic wave life cycles and full general circulation simulations have been successfully performed with a modified version of the European Centre for Medium-Range Weather Forecasts model.

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Steven Sandbach
,
John Thuburn
,
Danail Vassilev
, and
Michael G. Duda

Abstract

An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge–Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank–Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%–20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.

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