Parallel High-order Global Implicit Filter for the Spectral Element Dynamical Core on the Cubed Sphere

Parallel-friendly high-order global implicit filter for the spectral element dynamical core on the cubed sphere is developed to control small-scale numerical noises efficiently. The global discrete Laplacian operator matrix constructed by assembling the local discrete Laplacian operator matrix is used to make the Helmholtz-type global filter matrix. By preparing the inverted filter matrix in advance, only forward operation which is the multiplication of the inverted filter matrix and the global grid vector is required to carry out the filtering process. Also, the magnitude of inverted filter matrix components vary in significantly large range from (O-40) to (O -1) so that negligible matrix components less than a certain threshold value such as 10-15 or 10-10 can be removed reduce the operation count. It is found that the number of the matrix components after removal process is maintained with the help of diffusion coefficient and model time step size even though the horizontal resolution increases. This feature has advantages in parallelization of filtering process. Parallel scalability of filtering process only is nearly perfect to 512 CPUs. Scalability of the fully-equipped model is now under test and will be presented. Forward operating high-order filter is implemented in the spectral element hydrostatic dynamical core. Performance of forward operating high-order filter is evaluated using the dynamical core model intercomparison project (DCMIP) test cases and compared to explicit hyper-viscosity. Test case 2-0-0 which is steady-state atmosphere at rest in the presence of bottom topography is chosen, and we changed its moderately-steep orography to non-smoothed observed one to generate small-scale noise sufficiently. During 6 days of time integration, spurious flow was generated by the pressure gradient error. For a large time step size the model with explicit diffusion was found unstable, finally blowing-up at day 1. However, the model with second-order implicit filter was stable until the end of time integration. With a small time step size, the maximum wind speed for the implicit filter is lower and varies smoothly with time compare to explicit one. With a fourth-order implicit filter, trend of kinetic energy spectrum at day 6 is similar to spherical harmonics dynamical core with a fourth-order filter.