Paul A. Ullrich

A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid

A conservative multi-tracer transport algorithm on the cubed-sphere based on the semi-Lagrangian approach (CSLAM) has been developed. The scheme relies on backward trajectories and the resulting upstream cells (polygons) are approximated with great-circle arcs. Biquadratic polynomial functions are used for approximating the density distribution in the cubed-sphere grid cells. The upstream surface integrals associated with the conservative semi-Lagrangian scheme are computed as line-integrals by employing the Gauss-Green theorem. The line-integrals are evaluated using a combination of exact integrals and high-order Gaussian quadrature. The upstream cell (trajectories) information and computation of weights of integrals can be reused for each additional tracer. The CSLAM scheme is extensively tested with various standard benchmark test cases of solid-body rotation and deformational flow in both Cartesian and spherical geometry, and the results are compared with those of other published schemes. The CSLAM scheme is accurate, robust, and moreover, the edges and vertices of the cubed-sphere (discontinuities) do not affect the overall accuracy of the scheme. The CSLAM scheme exhibits excellent convergence properties and has an option for enforcing monotonicity. The advantages of introducing cross-terms in the fully two-dimensional biquadratic density distribution functions are also examined in the context of Cartesian as well as the cubed-sphere grid which has six local sub-domains with discontinuous edges and corners.

High-order finite-volume methods for the shallow-water equations on the sphere

This paper presents a third-order and fourth-order finite-volume method for solving the shallow-water equations on a non-orthogonal equiangular cubed-sphere grid. Such a grid is built upon an inflated cube placed inside a sphere and provides an almost uniform grid point distribution. The numerical schemes are based on a high-order variant of the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) pioneered by van Leer. In each cell the reconstructed left and right states are either obtained via a dimension-split piecewise-parabolic method or a piecewise-cubic reconstruction. The reconstructed states then serve as input to an approximate Riemann solver that determines the numerical fluxes at two Gaussian quadrature points along the cell boundary. The use of multiple quadrature points renders the resulting flux high-order. Three types of approximate Riemann solvers are compared, including the widely used solver of Rusanov, the solver of Roe and the new AUSM^+-up solver of Liou that has been designed for low-Mach number flows. Spatial discretizations are paired with either a third-order or fourth-order total-variation-diminishing Runge-Kutta timestepping scheme to match the order of the spatial discretization. The numerical schemes are evaluated with several standard shallow-water test cases that emphasize accuracy and conservation properties. These tests show that the AUSM^+-up flux provides the best overall accuracy, followed closely by the Roe solver. The Rusanov flux, with its simplicity, provides significantly larger errors by comparison. A brief discussion on extending the method to arbitrary order-of-accuracy is included.

Aquaplanet Experiments Using CAM’s Variable-Resolution Dynamical Core

AbstractA variable-resolution option has been added within the spectral element (SE) dynamical core of the U.S. Department of Energy (DOE)–NCAR Community Atmosphere Model (CAM). CAM-SE allows for static refinement via conforming quadrilateral meshes on the cubed sphere. This paper investigates the effect of mesh refinement in a climate model by running variable-resolution (var-res) simulations on an aquaplanet. The variable-resolution grid is a 2° (~222 km) grid with a refined patch of 0.25° (~28 km) resolution centered at the equator. Climatology statistics from these simulations are compared to globally uniform runs of 2° and 0.25°.A significant resolution dependence exists when using the CAM version 4 (CAM4) subgrid physical parameterization package across scales. Global cloud fraction decreases and equatorial precipitation increases with finer horizontal resolution, resulting in drastically different climates between the uniform grid runs and a physics-induced grid imprinting in the var-res simulation...

Operator-Split Runge–Kutta–Rosenbrock Methods for Nonhydrostatic Atmospheric Models

This paper presents a new approach for discretizing the nonhydrostatic Euler equations in Cartesian geometry using an operator-split time-stepping strategy and unstaggered upwind finite-volume model formulation. Following the method of lines, a spatial discretization of the governing equations leads to a set of coupled nonlinear ordinary differential equations. In general, explicit time-stepping methods cannot be applied directly to these equations because the large aspect ratio between the horizontal and vertical grid spacing leads to a stringent restriction on the time step to maintain numerical stability. Instead, an A-stable linearly implicit Rosenbrock method for evolving the vertical components of the equations coupled to atraditionalexplicitRunge‐Kutta formulainthehorizontalisproposed.Uptothird-ordertemporalaccuracy is achieved by carefully interleaving the explicit and linearly implicit steps. The time step for the resulting Runge‐Kutta‐Rosenbrock‐type semi-implicit method is then restricted only by the grid spacing and wave speed in the horizontal. The high-order finite-volume model is tested against a series of atmospheric flow problems to verify accuracy and consistency. The results of these tests reveal that this method is accurate, stable, and applicable to a wide range of atmospheric flows and scales.

MCore: A non-hydrostatic atmospheric dynamical core utilizing high-order finite-volume methods

This paper presents a new atmospheric dynamical core which uses a high-order upwind finite-volume scheme of Godunov type for discretizing the non-hydrostatic equations of motion on the sphere under the shallow-atmosphere approximation. The model is formulated on the cubed-sphere in order to avoid polar singularities. An operator-split Runge-Kutta-Rosenbrock scheme is used to couple the horizontally explicit and vertically implicit discretizations so as to maintain accuracy in time and space and enforce a global CFL condition which is only restricted by the horizontal grid spacing and wave speed. The Rosenbrock approach is linearly implicit and so requires only one matrix solve per column per time step. Using a modified version of the low-speed AUSM^+-up Riemann solver allows us to construct the vertical Jacobian matrix analytically, and so significantly improve the model efficiency. This model is tested against a series of typical atmospheric flow problems to verify accuracy and consistency. The test results reveal that this approach is stable, accurate and effective at maintaining sharp gradients in the flow.

Geometrically Exact Conservative Remapping (GECoRe): Regular Latitude–Longitude and Cubed-Sphere Grids

Abstract Land, ocean, and atmospheric models are often implemented on different spherical grids. As a conseqence coupling these model components requires state variables and fluxes to be regridded. For some variables, such as fluxes, it is paramount that the regridding algorithm is conservative (so that energy and water budget balances are maintained) and monotone (to prevent unphysical values). For global applications the cubed-sphere grids are gaining popularity in the atmospheric community whereas, for example, the land modeling groups are mostly using the regular latitude–longitude grid. Most existing regridding schemes fail to take advantage of geometrical symmetries between these grids and hence accuracy of the calculations can be lost. Hence, a new Geometrically Exact Conservative Remapping (GECoRe) scheme with a monotone option is proposed for remapping between regular latitude–longitude and gnomonic cubed-sphere grids. GECoRe is compared with existing remapping schemes published in the meteorolog...

Atmospheric Transport Schemes: Desirable Properties and a Semi-Lagrangian View on Finite-Volume Discretizations

This chapter has twofold purpose. After a short introduction to the mass continuity equations in atmospheric models, desirable properties for mass transport schemes intended for meteorological applications are discussed in some detail. This includes a discussion on the complications caused by the non-linearity of most problems of interest that makes it hard to define accuracy and convergence as the ‘truth’ is not known. Thereafter, some finite-volume schemes from the atmospheric literature are reviewed and discussed. To complement the large existing literature on finite-volume schemes, a less frequently discussed semi-Lagrangian derivation of the finite-volume method is given that focuses on ‘remap-type’ schemes where the space and time discretizations are combined rather than separated. A discussion on the challenges in deriving accurate schemes intended for global models and non-traditional spherical grids is given as well.

Characterizing Sierra Nevada Snowpack Using Variable-Resolution CESM

AbstractThe location, timing, and intermittency of precipitation in California make the state integrally reliant on winter-season snowpack accumulation to maintain its economic and agricultural livelihood. Of particular concern is that winter-season snowpack has shown a net decline across the western United States over the past 50 years, resulting in major uncertainty in water-resource management heading into the next century. Cutting-edge tools are available to help navigate and preemptively plan for these uncertainties. This paper uses a next-generation modeling technique—variable-resolution global climate modeling within the Community Earth System Model (VR-CESM)—at horizontal resolutions of 0.125° (14 km) and 0.25° (28 km). VR-CESM provides the means to include dynamically large-scale atmosphere–ocean drivers, to limit model bias, and to provide more accurate representations of regional topography while doing so in a more computationally efficient manner than can be achieved with conventional general ...

An evaluation of the variable resolution‐CESM for modeling California's climate

Author(s): Huang, X; Rhoades, AM; Ullrich, PA; Zarzycki, CM | Abstract:

Arbitrary-Order Conservative and Consistent Remapping and a Theory of Linear Maps: Part II

AbstractThis paper extends on the first part of this series by describing four examples of 2D linear maps that can be constructed in accordance with the theory of the earlier work. The focus is again on spherical geometry, although these techniques can be readily extended to arbitrary manifolds. The four maps include conservative, consistent, and (optionally) monotone linear maps (i) between two finite-volume meshes, (ii) from finite-volume to finite-element meshes using a projection-type approach, (iii) from finite-volume to finite-element meshes using volumetric integration, and (iv) between two finite-element meshes. Arbitrary order of accuracy is supported for each of the described nonmonotone maps.