2:15 PM - 2:30 PM
[MGI30-03] Development of an atmospheric nonhydrostatic dynamical core using the discontinuous Galerkin method: Consideration of topography
Keywords:High-resolution atmospheric simulation, Dynamical core, High-order numerical method, Topography
Introduction
Considering future high-resolution atmospheric simulations, we focus on the discontinuous Galerkin method (DGM), which is characterized by its simplicity and compactness for the high-order strategy. To investigate the suitability of DGM to the atmospheric simulations, we constructed a regional LES model based on DGM; we explored the degree (p) of the expansion polynomial required for atmospheric boundary layer turbulence and introduced a moist process. In those investigations, we did not consider topography. However, treatment of steep mountains is one of important issues in high-resolution experiments. A traditional method for handling the topography is to use a terrain-following coordinate (e.g., [1]). As a well-known problem of the terrain-following coordinate, when low-order discretization methods are used, the numerical errors in pressure gradient term can be large in the region with the topography, which results in spurious flows. On the other hand, in the case of high-order discretization methods, the influence of numerical errors related to the traditional terrain-following coordinate may not be essential because high-order methods have significantly small numerical errors for spatial scales larger than the effective resolution. To investigate the treatment of topography in higher-order DGM, we have extended our dynamics core. In this presentation, we show the preliminary results.
Formulation of dynamical core
The governing equations is a three-dimensional fully non-hydrostatic equations, and the vertical coordinates are formulated using general vertical coordinate. Here, we focus on the terrain-following coordinate. For the spatial discretization, nodal DGM (e.g., [2]) is applied. The computational domain is divided using hexahedral elements. The numerical integrals associated with the flux terms use the solution points. The fluxes at the element boundaries are evaluated using the Rusanov flux. The metric and transformed Jacobian associated with the vertical coordinate transformations are computed to p+1 accuracy in the framework of DGM.
Numerical experiment to validate our dynamical core
[Experimental setup] We conducted numerical experiments of a quasi-two-dimensional mountain wave based on [3]. The computational domain is 288 km horizontally and 30 km vertically. The topography is a bell-shaped mountain with a height of 1 m and a typical horizontal scale of 1 km. The initial condition is a stratified atmosphere with a horizontal uniform flow of 10 m/s (the Scorer number is 0.001). A periodic boundary condition is applied in the horizontal direction, and slip condition is imposed at the top and bottom of the model. A sponge layer is placed near the upper and horizontal boundaries. To compare with conventional atmospheric dynamical cores based on low-order grid-point methods, we also performed similar experiments using SCALE-RM ([4]) using a finite volume method (FVM) with totally second-order accuracy, but higher-order advection schemes can be used. As the advection scheme, we used the third- and seventh-order upwind schemes.
[Results] Figures (a) and (b) show the vertical wind distribution in the quasi-steady state. The spatial structure of mountain waves obtained from numerical experiments (colored lines) is qualitatively similar to a linear solution obtained from the stratified Boussinesq eqautaion (black dotted lines). The shading indicates the difference from the reference solution obtained from the high-resolution experiment. Compared with the DGM case at the same resolution, the result of SCALE-RM shows large phase errors of the wave and numerical errors associated with the vertical coordinate transformations. Figure (c) shows the resolution dependence of the L2 error norm calculated by the reference solution. The numerical convergence rate of SCALE-RM is about second-order accuracy, and the higher accuracy of the advection term does not contribute to the improvement of numerical errors in this test case. This indicates that the discretization accuracy of the pressure gradient term and metric influence the overall error. On the other hand, the errors of DGM with p=3, 7 is smaller than that of SCALE-RM when we compare the results in the similar resolution. Their numerical convergence rate is about 2nd to 4th-order accuracy. The numerical convergence rate of the DGM results for p=7 is much smaller than p+1 order. As a future task, we will investigate the reason.
Reference
[1] Gal-Chen and Somerville (1975), [2] Hesthaven & Warburton (2008), [3] Giraldo and Restelli (2008), [4] Nishizawa et al. (2015), Sato et al. (2015), [5] Smith (1980), Saito et al. (1998)
Considering future high-resolution atmospheric simulations, we focus on the discontinuous Galerkin method (DGM), which is characterized by its simplicity and compactness for the high-order strategy. To investigate the suitability of DGM to the atmospheric simulations, we constructed a regional LES model based on DGM; we explored the degree (p) of the expansion polynomial required for atmospheric boundary layer turbulence and introduced a moist process. In those investigations, we did not consider topography. However, treatment of steep mountains is one of important issues in high-resolution experiments. A traditional method for handling the topography is to use a terrain-following coordinate (e.g., [1]). As a well-known problem of the terrain-following coordinate, when low-order discretization methods are used, the numerical errors in pressure gradient term can be large in the region with the topography, which results in spurious flows. On the other hand, in the case of high-order discretization methods, the influence of numerical errors related to the traditional terrain-following coordinate may not be essential because high-order methods have significantly small numerical errors for spatial scales larger than the effective resolution. To investigate the treatment of topography in higher-order DGM, we have extended our dynamics core. In this presentation, we show the preliminary results.
Formulation of dynamical core
The governing equations is a three-dimensional fully non-hydrostatic equations, and the vertical coordinates are formulated using general vertical coordinate. Here, we focus on the terrain-following coordinate. For the spatial discretization, nodal DGM (e.g., [2]) is applied. The computational domain is divided using hexahedral elements. The numerical integrals associated with the flux terms use the solution points. The fluxes at the element boundaries are evaluated using the Rusanov flux. The metric and transformed Jacobian associated with the vertical coordinate transformations are computed to p+1 accuracy in the framework of DGM.
Numerical experiment to validate our dynamical core
[Experimental setup] We conducted numerical experiments of a quasi-two-dimensional mountain wave based on [3]. The computational domain is 288 km horizontally and 30 km vertically. The topography is a bell-shaped mountain with a height of 1 m and a typical horizontal scale of 1 km. The initial condition is a stratified atmosphere with a horizontal uniform flow of 10 m/s (the Scorer number is 0.001). A periodic boundary condition is applied in the horizontal direction, and slip condition is imposed at the top and bottom of the model. A sponge layer is placed near the upper and horizontal boundaries. To compare with conventional atmospheric dynamical cores based on low-order grid-point methods, we also performed similar experiments using SCALE-RM ([4]) using a finite volume method (FVM) with totally second-order accuracy, but higher-order advection schemes can be used. As the advection scheme, we used the third- and seventh-order upwind schemes.
[Results] Figures (a) and (b) show the vertical wind distribution in the quasi-steady state. The spatial structure of mountain waves obtained from numerical experiments (colored lines) is qualitatively similar to a linear solution obtained from the stratified Boussinesq eqautaion (black dotted lines). The shading indicates the difference from the reference solution obtained from the high-resolution experiment. Compared with the DGM case at the same resolution, the result of SCALE-RM shows large phase errors of the wave and numerical errors associated with the vertical coordinate transformations. Figure (c) shows the resolution dependence of the L2 error norm calculated by the reference solution. The numerical convergence rate of SCALE-RM is about second-order accuracy, and the higher accuracy of the advection term does not contribute to the improvement of numerical errors in this test case. This indicates that the discretization accuracy of the pressure gradient term and metric influence the overall error. On the other hand, the errors of DGM with p=3, 7 is smaller than that of SCALE-RM when we compare the results in the similar resolution. Their numerical convergence rate is about 2nd to 4th-order accuracy. The numerical convergence rate of the DGM results for p=7 is much smaller than p+1 order. As a future task, we will investigate the reason.
Reference
[1] Gal-Chen and Somerville (1975), [2] Hesthaven & Warburton (2008), [3] Giraldo and Restelli (2008), [4] Nishizawa et al. (2015), Sato et al. (2015), [5] Smith (1980), Saito et al. (1998)