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Peter H. Lauritzen, Eigil Kaas, and Bennert Machenhauer

Abstract

A locally mass conservative shallow-water model using a two-time-level, semi-implicit, semi-Lagrangian integration scheme is presented. The momentum equations are solved with the traditional semi-Lagrangian gridpoint form. The explicit continuity equation is solved using a cell-integrated semi-Lagrangian scheme, and the semi-implicit part is designed such that the resulting elliptic equation is on the same form as for the traditional semi-Lagrangian gridpoint system.

The accuracy of the model is assessed by running standard test cases adapted to a limited-area domain. The accuracy and efficiency of the new model is comparable to traditional semi-Lagrangian methods.

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Peter H. Lauritzen and Ramachandran D. Nair

Abstract

A high-order monotone and conservative cascade remapping algorithm between spherical grids (CaRS) is developed. This algorithm is specifically designed to remap between the cubed-sphere and regular latitude–longitude grids. The remapping approach is based on the conservative cascade method in which a two-dimensional remapping problem is split into two one-dimensional problems. This allows for easy implementation of high-order subgrid-cell reconstructions as well as the application of advanced monotone filters. The accuracy of CaRS is assessed by remapping analytic fields from the regular latitude–longitude grid to the cubed-sphere grid. In terms of standard error measures, CaRS is found to be competitive relative to an existing algorithm when regridding from a fine to a coarse grid and more accurate when regridding from a coarse to a fine grid.

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Christoph Erath, Peter H. Lauritzen, and Henry M. Tufo

Abstract

It is the purpose of this short article to analyze mass conservation in high-order rigorous remapping schemes, which contrary to flux-based methods, relies on elaborate integral constraints over overlap areas and reconstruction functions. For applications on the sphere these integral constraints may be violated primarily as a result of inexact or ill-conditioned integration and the authors propose a generic, local, and multitracer efficient method that guarantees that the integral constraints are satisfied in discrete space irrespective of the accuracy of the numerical integration method and slight inaccuracies in the computation of overlap areas. The authors refer to this method as enforcement of consistency as it is based on integral constraints valid in continuous space. The consistency enforcement method is illustrated in idealized transport tests with the Conservative Semi-Lagrangian Multitracer scheme (CSLAM) in the High Order Method Modeling Environment (HOMME) where the analytic integrals, which were found to be ill conditioned at certain resolutions and flow conditions, have been replaced with robust quadrature. This violates mass conservation; however, with the consistency enforcement method, mass conservation is inherent even with low-order quadrature and renders rigorous remap schemes such as CSLAM (which was previously limited to gnomonic cubed-sphere grids) mass conservative on any spherical grid.

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Paul A. Ullrich, Peter H. Lauritzen, and Christiane Jablonowski

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 meteorological literature. It is concluded here that the geometrically exact scheme significantly improves the accuracy of the resulting remapping in idealized test cases.

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Aman Gupta, Edwin P. Gerber, R. Alan Plumb, and Peter H. Lauritzen

Abstract

Accurate representation of stratospheric trace gas transport is important for ozone modeling and climate projection. Intermodel spread can arise from differences in the representation of transport by the diabatic (overturning) circulation versus comparatively faster adiabatic mixing by breaking waves, or through numerical errors, primarily diffusion. This study investigates the impact of these processes on transport using an idealized tracer, the age of air. Transport is assessed in two state-of-the-art dynamical cores based on fundamentally different numerical formulations: finite-volume and spectral element. Integrating the models in free-running and nudged tropical wind configurations reveals the crucial impact of tropical dynamics on stratospheric transport. Using age-budget theory, vertical and horizontal gradients of age allow comparison of the roles of the diabatic circulation, adiabatic mixing, and the numerical diffusive flux. Their respective contribution is quantified by connecting the full 3D model to the tropical leaky pipe framework of Neu and Plumb. Transport by the two cores varies significantly in the free-running integrations, with the age in the middle stratosphere differing by about 2 years primarily due to differences in adiabatic mixing. When winds in the tropics are constrained, the difference in age drops to about 0.5 years; in this configuration, more than half the difference is due to the representation of the diabatic circulation. Numerical diffusion is very sensitive to the resolution of the core, but does not play a significant role in differences between the cores when they are run at comparable resolution. It is concluded that fundamental differences rooted in dynamical core formulation can account for a substantial fraction of transport bias between climate models.

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May Wong, William C. Skamarock, Peter H. Lauritzen, Joseph B. Klemp, and Roland B. Stull

Abstract

A recently developed cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic atmospheric solver that uses the conservative semi-Lagrangian multitracer (CSLAM) transport scheme is extended to include orographic influences. With the introduction of a new semi-implicit CISL discretization of the continuity equation, the nonhydrostatic solver, called CSLAM-NH, has been shown to ensure inherently conservative and numerically consistent transport of air mass and other scalar variables, such as moisture and passive tracers. The extended CSLAM-NH presented here includes two main modifications: transformation of the equation set to a terrain-following height coordinate to incorporate orography and an iterative centered-implicit time-stepping scheme to enhance the stability of the scheme associated with gravity wave propagation at large time steps. CSLAM-NH is tested for a suite of idealized 2D flows, including linear mountain waves (dry), a downslope windstorm (dry), and orographic cloud formation.

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Jared P. Whitehead, Christiane Jablonowski, Richard B. Rood, and Peter H. Lauritzen

Abstract

The dynamical core of an atmospheric general circulation model is engineered to satisfy a delicate balance between numerical stability, computational cost, and an accurate representation of the equations of motion. It generally contains either explicitly added or inherent numerical diffusion mechanisms to control the buildup of energy or enstrophy at the smallest scales. The diffusion fosters computational stability and is sometimes also viewed as a substitute for unresolved subgrid-scale processes. A particular form of explicitly added diffusion is horizontal divergence damping.

In this paper a von Neumann stability analysis of horizontal divergence damping on a latitude–longitude grid is performed. Stability restrictions are derived for the damping coefficients of both second- and fourth-order divergence damping. The accuracy of the theoretical analysis is verified through the use of idealized dynamical core test cases that include the simulation of gravity waves and a baroclinic wave. The tests are applied to the finite-volume dynamical core of NCAR’s Community Atmosphere Model version 5 (CAM5). Investigation of the amplification factor for the divergence damping mechanisms explains how small-scale meridional waves found in a baroclinic wave test case are not eliminated by the damping.

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May Wong, William C. Skamarock, Peter H. Lauritzen, and Roland B. Stull

Abstract

A Cartesian semi-implicit solver using the Conservative Semi-Lagrangian Multitracer (CSLAM) transport scheme is constructed and tested for shallow-water (SW) flows. The SW equations solver (CSLAM-SW) uses a discrete semi-implicit continuity equation specifically designed to ensure a conservative and consistent transport of constituents by avoiding the use of a constant mean reference state. The algorithm is constructed to be similar to typical conservative semi-Lagrangian semi-implicit schemes, requiring at each time step a single linear Helmholtz equation solution and a single application of CSLAM. The accuracy and stability of the solver is tested using four test cases for a radially propagating gravity wave and two barotropically unstable jets. In a consistency test using the new solver, the specific concentration constancy is preserved up to machine roundoff, whereas a typical formulation can have errors many orders of magnitude larger. In addition to mass conservation and consistency, CSLAM-SW also ensures shape preservation by combining the new scheme with existing shape-preserving filters. With promising SW test results, CSLAM-SW shows potential for extension to a nonhydrostatic, fully compressible system solver for numerical weather prediction and climate models.

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May Wong, William C. Skamarock, Peter H. Lauritzen, Joseph B. Klemp, and Roland B. Stull

Abstract

A cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic solver for the fully compressible moist Euler equations in two-dimensional Cartesian (xz) geometry is presented. The semi-implicit CISL solver uses the inherently conservative semi-Lagrangian multitracer transport scheme (CSLAM) and a new flux-form semi-implicit formulation of the continuity equation that ensures numerically consistent transport. The flux-form semi-implicit formulation is based on a recent successful approach in a shallow-water equations (SWE) solver (CSLAM-SW). With the new approach, the CISL semi-implicit nonhydrostatic solver (CSLAM-NH) is able to ensure conservative and consistent transport by avoiding the need for a time-independent mean reference state. Like its SWE counterpart, the nonhydrostatic solver presented here is designed to be similar to typical semi-Lagrangian semi-implicit schemes, such that only a single linear Helmholtz equation solution and a single call to CSLAM are required per time step. To demonstrate its stability and accuracy, the solver is applied to a set of three idealized test cases: a density current (dry), a gravity wave (dry), and a squall line (moist). A fourth test case shows that shape preservation of passive tracers is ensured by coupling the semi-implicit CISL formulation with existing shape-preserving filters. Results show that CSLAM-NH solutions compare well with other existing solvers for the three test cases, and that it is shape preserving.

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Kevin A. Reed, Brian Medeiros, Julio T. Bacmeister, and Peter H. Lauritzen

Abstract

In the continued effort to understand the climate system and improve its representation in atmospheric general circulation models (AGCMs), it is crucial to develop reduced-complexity frameworks to evaluate these models. This is especially true as the AGCM community advances toward high horizontal resolutions (i.e., grid spacing less than 50 km), which will require interpreting and improving the performance of many model components. A simplified global radiative–convective equilibrium (RCE) configuration is proposed to explore the implication of horizontal resolution on equilibrium climate. RCE is the statistical equilibrium in which the radiative cooling of the atmosphere is balanced by heating due to convection.

In this work, the Community Atmosphere Model, version 5 (CAM5), is configured in RCE to better understand tropical climate and extremes. The RCE setup consists of an ocean-covered Earth with diurnally varying, spatially uniform insolation and no rotation effects. CAM5 is run at two horizontal resolutions: a standard resolution of approximately 100-km grid spacing and a high resolution of approximately 25-km spacing. Surface temperature effects are considered by comparing simulations using fixed, uniform sea surface temperature with simulations using an interactive slab-ocean model. The various CAM5 configurations provide useful insights into the simulation of tropical climate as well as the model’s ability to simulate extreme precipitation events. In particular, the manner in which convection organizes is shown to be dependent on model resolution and the surface configuration (including surface temperature), as evident by differences in cloud structure, circulation, and precipitation intensity.

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