A mass and energy conserving spectral element atmospheric dynamical core on the cubed-sphere grid

Abstract

Abstract. We present results from a conservative formulation of the spectral element method applied to global atmospheric circulation modeling. Exact local conservation of both mass and energy is obtained via a new compatible formulation of the spectral element method. Compatibility insures that the key integral property of the divergence and gradient operators required to show conservation also hold in discrete form. The spectral element method is used on a cubed-sphere grid to discretize the horizontal directions on the sphere. It can be coupled to any conservative vertical/radial discretization. The accuracy and conservation properties of the method are illustrated using a baroclinic instability test case. IntroductionThe spectral element method is a finite element method which relies on polynomial basis functions and quadrilateral elements. The equations of interest are solved in integral formulation and the integrals are evaluated with Gauss-Lobatto quadrature within each element. The GaussLobatto quadrature leads to a diagonal mass matrix, which allows the method to obtain spectral accuracy while retaining both parallel efficiency and the geometric flexibility of unstructured finite elements grids. The method has proven accurate and efficient for a wide variety of geophysical problems, including global atmospheric circulation modeling [1,2,3,4] ocean modeling [5,6,7] and planetary scale seismology [8]. The method has unsurpassed parallel performance; it was used for earthquake modeling by the 2003 Gordon Bell Best Performance winner [8] and for climate modeling by a 2002 Gordon Bell Award honorable mention [9].In this work, we use a modified version of the spectral element component of HOMME, the High Order Multiscale Modeling Environment [10], which we refer to as HOMME-SE. HOMME-SE models the global circulation of the Earth's atmosphere using the three-dimensional hydrostatic primitive equations. The formulation of the equations is taken from [11]. Spectral elements on a cubed-sphere grid are used to discretize the horizontal directions (the surface of the Earth). In the radial direction, HOMME-SE uses the hybrid η pressure vertical coordinate system [12,11]. We made several modifications to HOMME-SE in order to make the method locally conservative: we implemented the compatible spectral element formulation from [13], which involved slightly different forms of the metric terms in the discrete integrals, divergence, gradient and vorticity operators, and we switched from advection of log(p s ) to surface pressure p s .

Performance of the HOMME dynamical core in the aqua-planet configuration of NCAR CAM4: equatorial waves

“…This method is designed for fully unstructured quadrilateral meshes. The current configurations in CAM are based on the cubed-sphere grid (Taylor et al, 2007(Taylor et al, , 2008Bhanot et al, 2008;Nair et al, 2009). Spectral elements are a kind of a continuous Galerkin hp finite element method (Karniadakis and Sherwin, 1999;Canuto et al, 2007), where h is the number of elements and p the polynomial order.…”

“…Scalable performance is necessary to efficiently exploit the massively-parallel petascale systems that will dominate high-performance computing for the foreseeable future (Nair and Tufo, 2007;Taylor et al, 2008;Nair et al, 2009). The High Order Method Modeling Environment (HOMME) is a framework developed at NCAR (Dennis et al, 2005;Taylor et al, 2007Taylor et al, , 2008Taylor, 2010) to investigate using high-order element-based methods to build scalable, accurate and conservative atmospheric general circulation models (AGCM). The primary object of HOMME project is to provide the atmospheric science community a framework for building the next generation of AGCMs based on high-order numerical methods that efficiently scale to hundred-of-thousands of processors, achieve scientifically useful integration rates, provide monotonic and mass conserving transport of multiple species, and can easily be coupled to community physics packages (Taylor, 2010).…”

Progress Report 2008: A Scalable and Extensible Earth System Model for Climate Change Science

“…1 The numerics in HOMME did not conserve mass or energy, which we addressed by developing a new compatible formulation of spectral elements (Taylor et al, 2007), making HOMME the first dynamical core in the CCSM to conserve both mass and energy without the use of ad-hoc fixers. 2 The original limiter based dissipation mechanisms in HOMME resulted in noticeable grid imprinting in the solution, which we eliminated by replacing the limiters with an isotropic hyper-viscosity operator.…”

Mass conservation of the unified continuous and discontinuous element-based Galerkin methods on dynamically adaptive grids with application to atmospheric simulations

“…None of the aforementioned papers report a quantitative measure of the conservation of mass, energy, or any quantity for that matter. [12] (with a later follow-up in [13]) addressed the conservation issue of the conforming continuous Galerkin method on the cubed-sphere grid showing the mimetic properties of the HOMME-SE model and extends [10] to inexact integration used in the model. We address the question how non-conforming adaptive mesh refinement affects the mass conservation of both the CG and DG methods.…”