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William H. Lipscomb
and
Todd D. Ringler

Abstract

Weather and climate models contain equations for transporting conserved quantities such as the mass of air, water, ice, and associated tracers. Ideally, the numerical schemes used to solve these equations should be conservative, spatially accurate, and monotonicity-preserving. One such scheme is incremental remapping, previously developed for transport on quadrilateral grids. Here the incremental remapping scheme is reformulated for a spherical geodesic grid whose cells are hexagons and pentagons. The scheme is tested in a shallow-water model with both uniform and varying velocity fields. Solutions for standard shallow-water test cases 1, 2, and 5 are obtained with a centered scheme, a flux-corrected transport (FCT) scheme, and the remapping scheme. The three schemes are about equally accurate for transport of the height field. For tracer transport, remapping is far superior to the centered scheme, which produces large overshoots, and is generally smoother and more accurate than FCT. Remapping has a high startup cost associated with geometry calculations but is nearly twice as fast as FCT for each added tracer. As a result, remapping is cheaper than FCT for transport of more than about seven tracers.

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Todd D. Ringler
and
David A. Randall

Abstract

Using the shallow water equations, a numerical framework on a spherical geodesic grid that conserves domain-integrated mass, potential vorticity, potential enstrophy, and total energy is developed. The numerical scheme is equally applicable to hexagonal grids on a plane and to spherical geodesic grids. This new numerical scheme is compared to its predecessor and it is shown that the new scheme does considerably better in conserving potential enstrophy and energy. Furthermore, in a simulation of geostrophic turbulence, the new numerical scheme produces energy and enstrophy spectra with slopes of approximately K −3 and K −1, respectively, where K is the total wavenumber. These slopes are in agreement with theoretical predictions. This work also exhibits a discrete momentum equation that is compatible with the Z-grid vorticity-divergence equation.

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Todd D. Ringler
and
David A. Randall

Abstract

Shallow-water equations discretized on a perfect hexagonal grid are analyzed using both a momentum formulation and a vorticity-divergence formulation. The vorticity-divergence formulation uses the unstaggered Z grid that places mass, vorticity, and divergence at the centers of the hexagons. The momentum formulation uses the staggered ZM grid that places mass at the centers of the hexagons and velocity at the corners of the hexagons. It is found that the Z grid and the ZM grid are identical in their simulation of the physical modes relevant to geostrophic adjustment. Consistent with the continuous system, the simulated inertia–gravity wave phase speeds increase monotonically with increasing total wavenumber and, thus, all waves have nonzero group velocities.

Since a grid of hexagons has twice as many corners as it has centers, the ZM grid has twice as many velocity points as it has mass points. As a result, the ZM-grid velocity field is discretized at a higher resolution than the mass field and, therefore, resolves a larger region of wavenumber space than the mass field. We solve the ∇2 f = λf eigenvalue problem with periodic boundary conditions on both the Z grid and ZM grid to determine the modes that can exist on each grid. The mismatch between mass and momentum leads to computational modes in the velocity field. Two techniques that can be used to control these computational modes are discussed. One technique is to use a dissipation operator that captures or “sees” the smallest-scale variations in the velocity field. The other technique is to invert elliptic equations in order to filter the high wavenumber part of the momentum field.

Results presented here lead to the conclusion that the ZM grid is an attractive alternative to the Z grid, and might be particularly useful for ocean modeling.

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Sara A. Rauscher
and
Todd D. Ringler

Abstract

The effects of a variable-resolution mesh on simulated midlatitude baroclinic eddies in idealized settings are examined. Both aquaplanet and Held–Suarez experiments are performed using the Model for Prediction Across Scales-Atmosphere (MPAS-A) hydrostatic dynamical core implemented within the National Science Foundation–Department of Energy (NSF–DOE) Community Atmosphere Model (CAM-MPAS-A). In the real world, midlatitude eddy activity is organized by orography, land–sea contrasts, and sea surface temperature anomalies. In these zonally symmetric idealized settings, transients should have an equal probability of occurring at any longitude. However, the use of a variable-resolution mesh with a circular high-resolution region centered at 30°N results in a maximum in eddy kinetic energy on the eastern side and downstream of this high-resolution region in both aquaplanet and Held–Suarez CAM-MPAS-A simulations. The presence of a geographically confined maximum in both simulations suggests this response is mainly attributable to CAM-MPAS-A’s ability to resolve eddies via the model dynamics as resolution increases. However, in the aquaplanet simulation, a secondary maximum in eddy kinetic energy is present, which is probably linked to the resolution dependencies of the CAM physics. These mesh responses must be considered when interpreting real-world variable-resolution CAM-MPAS-A simulations, particularly in climate change experiments.

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Todd D. Ringler
,
Ross P. Heikes
, and
David A. Randall

Abstract

This paper documents the development and testing of a new type of atmospheric dynamical core. The model solves the vorticity and divergence equations in place of the momentum equation. The model is discretized in the horizontal using a geodesic grid that is nearly uniform over the entire globe. The geodesic grid is formed by recursively bisecting the triangular faces of a regular icosahedron and projecting those new vertices onto the surface of the sphere. All of the analytic horizontal operators are reduced to line integrals, which are numerically evaluated with second-order accuracy. In the vertical direction the model can use a variety of coordinate systems, including a generalized sigma coordinate that is attached to the top of the boundary layer. Terms related to gravity wave propagation are isolated and an efficient semi-implicit time-stepping scheme is implemented. Since this model combines many of the positive attributes of both spectral models and conventional finite-difference models into a single dynamical core, it represents a distinctively new approach to modeling the atmosphere’s general circulation.

The model is tested using the idealized forcing proposed by Held and Suarez. Results are presented for simulations using 2562 polygons (approximately 4.5° × 4.5°) and using 10 242 polygons (approximately 2.25° × 2.25°). The results are compared to those obtained with spectral model simulations truncated at T30 and T63. In terms of first and second moments of state variables such as the zonal wind, meridional wind, and temperature, the geodesic grid model results using 2562 polygons are comparable to those of a spectral model truncated at slightly less than T30, while a simulation with 10 242 polygons is comparable to a spectral model simulation truncated at slightly less than T63.

In order to further demonstrate the viability of this modeling approach, preliminary results obtained from a full-physics general circulation model that uses this dynamical core are presented. The dominant features of the DJF climate are captured in the full-physics simulation.

In terms of computational efficiency, the geodesic grid model is somewhat slower than the spectral model used for comparison. Model timings completed on an SGI Origin 2000 indicate that the geodesic grid model with 10 242 polygons is 20% slower than the spectral model truncated at T63. The geodesic grid model is more competitive at higher resolution than at lower resolution, so further optimization and future trends toward higher resolution should benefit the geodesic grid model.

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Todd D. Ringler
,
Doug Jacobsen
,
Max Gunzburger
,
Lili Ju
,
Michael Duda
, and
William Skamarock

Abstract

The ability to solve the global shallow-water equations with a conforming, variable-resolution mesh is evaluated using standard shallow-water test cases. While the long-term motivation for this study is the creation of a global climate modeling framework capable of resolving different spatial and temporal scales in different regions, the process begins with an analysis of the shallow-water system in order to better understand the strengths and weaknesses of the approach developed herein. The multiresolution meshes are spherical centroidal Voronoi tessellations where a single, user-supplied density function determines the region(s) of fine- and coarse-mesh resolution. The shallow-water system is explored with a suite of meshes ranging from quasi-uniform resolution meshes, where the grid spacing is globally uniform, to highly variable resolution meshes, where the grid spacing varies by a factor of 16 between the fine and coarse regions. The potential vorticity is found to be conserved to within machine precision and the total available energy is conserved to within a time-truncation error. This result holds for the full suite of meshes, ranging from quasi-uniform resolution and highly variable resolution meshes. Based on shallow-water test cases 2 and 5, the primary conclusion of this study is that solution error is controlled primarily by the grid resolution in the coarsest part of the model domain. This conclusion is consistent with results obtained by others. When these variable-resolution meshes are used for the simulation of an unstable zonal jet, the core features of the growing instability are found to be largely unchanged as the variation in the mesh resolution increases. The main differences between the simulations occur outside the region of mesh refinement and these differences are attributed to the additional truncation error that accompanies increases in grid spacing. Overall, the results demonstrate support for this approach as a path toward multiresolution climate system modeling.

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William C. Skamarock
,
Joseph B. Klemp
,
Michael G. Duda
,
Laura D. Fowler
,
Sang-Hun Park
, and
Todd D. Ringler

Abstract

The formulation of a fully compressible nonhydrostatic atmospheric model called the Model for Prediction Across Scales–Atmosphere (MPAS-A) is described. The solver is discretized using centroidal Voronoi meshes and a C-grid staggering of the prognostic variables, and it incorporates a split-explicit time-integration technique used in many existing nonhydrostatic meso- and cloud-scale models. MPAS can be applied to the globe, over limited areas of the globe, and on Cartesian planes. The Voronoi meshes are unstructured grids that permit variable horizontal resolution. These meshes allow for applications beyond uniform-resolution NWP and climate prediction, in particular allowing embedded high-resolution regions to be used for regional NWP and regional climate applications. The rationales for aspects of this formulation are discussed, and results from tests for nonhydrostatic flows on Cartesian planes and for large-scale flow on the sphere are presented. The results indicate that the solver is as accurate as existing nonhydrostatic solvers for nonhydrostatic-scale flows, and has accuracy comparable to existing global models using icosahedral (hexagonal) meshes for large-scale flows in idealized tests. Preliminary full-physics forecast results indicate that the solver formulation is robust and that the variable-resolution-mesh solutions are well resolved and exhibit no obvious problems in the mesh-transition zones.

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