High Order Method Modeling Environment

Model dynamics test
Simulation of the 3D non-hydrostatic gravity wave propagation (a DCMIP benchmark test) on a reduced planet using the HOMAM dynamical core. This test examines model response to short-time-scale wave motion triggered by a local perturbation, where the initial state is hydrostatically balanced. The upper panel (2D vertical slice) shows the initial potential temperature perturbation, which triggers the evolution of gravity waves as shown in the lower panel after 3600 seconds.

The High-Order Method Modeling Environment (HOMME) is a hydrostatic framework to investigate using high-order element-based methods to build conservative and accurate dynamical cores. Currently, HOMME employs the Spectral Element (SE) and Discontinuous Galerkin (DG) methods on a cubed-sphere tiled with quadrilateral elements. HOMME can be configured to solve the shallow water or the dry/moist primitive equations with explicit time-stepping. The objective of this project is to extend HOMME to a framework capable of providing the atmospheric science community with a new generation of atmospheric general circulation models (AGCMs) for the CESM (Community Earth System Model). Currently the SE version of HOMME is the default dynamical core for NCAR’s Community Atmosphere Model (CAM), and HOMME-SE can efficiently scale hundreds of thousands processors on a supercomputer. With the emergence of petascale computing resources, it is now possible to develop high-resolution (cloud-resolving) global models at non-hydrostatic (NH) scales.

In FY2014, a major research focus was to develop a 2D NH model based on DG methods to facilitate testing various time-stepping approaches and the vertical height-based coordinates, which would be applicable to the NH extension of HOMME. The time-split approach, known as the “horizontally explicit and vertically implicit” (HEVI) scheme, was found to be effective on the DG-NH model (Bao et al., 2015; DOI:10.1175/MWR-D-14-00083.1). The maximum stable time step for explicit time discretizations is dictated by the Courant–Friedrichs–Lewy (CFL) stability limit. At a higher resolution (smaller grid spacing), the CFL limit requires extremely smaller time steps, and is not practical for global NH models simulating climate. However, the stringent CFL limit associated with vertical high resolution can be remedied by using a dimension-splitting procedure that treats the vertical component of the equations in an implicit manner and the horizontal components explicitly, and the HEVI scheme does it precisely.

In FY2015, the HOMME framework has been extended to a non-hydrostatic dynamical core named the “High-Order Multiscale Atmospheric Model (HOMAM).” The horizontal aspects of discretization remain the same as that of the current HOMME cubed-sphere grid system, and discontinuous Galerkin (DG) spatial discretization is used. The DG method possesses computationally desirable properties such as local and global conservation, geometric flexibility, high on-processor operations, and minimal communication footprints. The vertical discretization is based on terrain-following z-coordinates as used in the 2D DG-NH model.

Several measures have been taken to extend the HOMME framework in preparation for the NH model implementation. These include restructuring the code, improving the efficiency of the DG algorithms, enhancing the parallel communication, and incorporating various DCMIP (Dynamical Core Model Intercomparison Project) benchmark tests to validate the new 3D NH model formulation. The time-stepping schemes HEVI and HEVE (horizontally explicit and vertically explicit) have been implemented in HOMAM, and their performance has been compared with a fully explicit Runge-Kutta (RK) method.

Accuracy test
Convergence of error for a DCMIP benchmark 3D advection test after one-day simulation with varying vertical and horizontal grid resolutions. HOMAM employs several time-stepping methods including HEVI, HEVE, and fully explicit RK method. The convergence rate shows a second-order accuracy as expected for a time-split model.

The figure at right shows the convergence of error norms for different time-stepping schemes as used in HOMAM with a DCMIP benchmark experiment known as the “Hadley test.” This test employs a smooth deformational flow that mimics a Hadley-like meridional circulation, and analytic solution is available at the end of the one-day simulation for comparison. This test is designed to investigate the impact of horizontal-vertical spatial splitting on the accuracy of the scheme. With HEVI and HEVE time integration methods, the HOMAM convergence results show a second-order accuracy, irrespective of a particular time integrator, which is acceptable for a time-splitting model (Nair et al., 2015; DOI:10.1016/j.procs.2015.05.471).

The figure at top shows the HOMAM simulation results with the NH gravity wave test as recommended in the DCMIP test suite. This is an idealized test involving full 3D nonlinear dynamics, where the initial state is hydrostatically balanced. This test examines the response of a model to short-time-scale wave motion triggered by a local perturbation and provides an excellent tool to test model dynamics. For this test, an overlaid potential temperature perturbation triggers the evolution of gravity waves up to a 1-hour period on a reduced (shrunken) planet. The preliminary result with HOMAM is encouraging, and comparable to that of the DCMIP benchmark results produced with established models.

This work supports CISL’s science imperative to develop mathematical research codes that improve modeling. Specifically, it fulfills the strategic action item to further develop the HOMME dynamical core. Primary support for HOMME is provided by NSF Core funding. This project was partially funded by the Korean Institute of Atmospheric Prediction Systems of Seoul, South Korea.