9:30 AM - 9:45 AM
[MGI35-03] A large-eddy simulation of atmospheric boundary layer turbulence using the discontinuous Galerkin method
Keywords:atmospheric boundary layer, large eddy simulation, high-order numerical scheme
1. Introduction
Recently, spatial resolution of global atmospheric models has been approaching the spatial scales targeted in large-eddy simulations (LES). One of the challenges toward global LES is the numerical accuracy of the dynamical process. Focusing on the atmospheric boundary layer, Kawai and Tomita (2020) discussed the order of accuracy required for discretization errors of the advection terms not to dominate the eddy viscosity term by the turbulence model in the framework of the finite difference method (FDM). The derived criteria suggest that at least seventh- or eighth-order accuracy is required. The next issues are to verify the effect of increasing numerical accuracy of all terms, and to attempt the dynamics-physics coupling considered the effective resolution of the dynamics. In addition, we need to explore numerical schemes providing the accurate representation of the physical field and highly computational efficiency when considering large-scale parallel computers. The discontinuous Galerkin method (DGM) may be useful for these issues. The strategy of high-order discretization is straightforward and compact. In terms of the dynamics-physics coupling, many degrees of freedom within the element may give us some knowledges for utilizing accurate dynamical fields to calculate the physical processes. In this study, to investigate the applicability of DGM to global LES, we constructed a limited-area atmospheric LES model using DGM, and conducted numerical experiments. Here, we present the preliminary results.
2. LES of atmospheric boundary layer turbulence using DGM
2.1 Numerical method
The governing equations of dynamics is the fully compressible equations, and the turbulent process is represented by a Smagorinsky-Lilly type model (e.g., Brown et al., 1994). The spatial discretization is based on the nodal DGM (e.g., Hesthaven & Warburton, 2008). To archive the eight-th order accuracy in terms of truncation errors, we use the hexahedral finite element with 83 nodes. As the numerical flux, we adopt the Rusanov flux for inviscid terms and the central flux for eddy viscous terms. As the spatial filter in LES, we use the16th-order elementwise modal filter with the exponential form. This filter rapidly removes the fluid structure whose wavelength is near two-grids. For this case, although the definition of spatial filter length seems to be not obvious, here we consider the filter length is essentially located between two- and four-grids.
2.2 Experimental setup
We conduct numerical experiments of idealized atmospheric boundary layer turbulence based on Nishizawa et al. (2015). The domain size is 9.6x9.6x3 km3. Initially, the horizontal wind is 5m/s, and the perturbation of potential temperature is added into stably stratified atmosphere. The constant heat flux with 200 W/m2 is injected at the surface. We perform the integration over four hours. The size of finite element is set to 200 m, which means the corresponding grid size in FDM (Δx) is approximately 30 m. To obtain the knowledge of the dynamic-physics coupling in DGM, we investigate how sensitive is the intensity of the modal filter (represented by the decay time for maximum wavenumber, Tnd) and the spatial filter length for turbulent model (Δsgs) to the energy spectra. We set these parameters such that Tnd/Δt~0.1, 10, 100 (where Δt is the time step), and Δsgs/Δx=1, 2, 4. Here, we regard the case of Tnd/Δt~10 and Δsgs/Δx=2 as the control experiment.
2.3 Results
We show the energy spectra of three-dimensional winds in the figure. For comparison, the results of SCALE-RM using the conservative FDM are also shown by gray lines. The slope for the control experiment becomes steeper than that of the -5/3 power law at wavelengths shorter than 200 m. Over the wavelength between 500 m and 200 m, the spectra for the control experiment and the stronger modal filter case has a little shallower slope than that of the -5/3 power law. This indicates the accumulation of energy. The sensitivity of the spatial filter length is higher than that of the modal filter’s intensity. In terms of the dynamics-physics coupling considering the effective resolution of dynamics, we expected that Δsgs/Δx= 4 would be appropriate in this experiment, however the actual spectra for Δsgs/Δx= 4 show too strong dissipation and deviate significantly from the -5/3 power law. To consider the reasons for the above results, we will revisit the formulation of LES in DGM. We will also conduct additional experiments with the higher resolution to make precise discussion, because the resolved inertial subrange is slightly narrow.
Recently, spatial resolution of global atmospheric models has been approaching the spatial scales targeted in large-eddy simulations (LES). One of the challenges toward global LES is the numerical accuracy of the dynamical process. Focusing on the atmospheric boundary layer, Kawai and Tomita (2020) discussed the order of accuracy required for discretization errors of the advection terms not to dominate the eddy viscosity term by the turbulence model in the framework of the finite difference method (FDM). The derived criteria suggest that at least seventh- or eighth-order accuracy is required. The next issues are to verify the effect of increasing numerical accuracy of all terms, and to attempt the dynamics-physics coupling considered the effective resolution of the dynamics. In addition, we need to explore numerical schemes providing the accurate representation of the physical field and highly computational efficiency when considering large-scale parallel computers. The discontinuous Galerkin method (DGM) may be useful for these issues. The strategy of high-order discretization is straightforward and compact. In terms of the dynamics-physics coupling, many degrees of freedom within the element may give us some knowledges for utilizing accurate dynamical fields to calculate the physical processes. In this study, to investigate the applicability of DGM to global LES, we constructed a limited-area atmospheric LES model using DGM, and conducted numerical experiments. Here, we present the preliminary results.
2. LES of atmospheric boundary layer turbulence using DGM
2.1 Numerical method
The governing equations of dynamics is the fully compressible equations, and the turbulent process is represented by a Smagorinsky-Lilly type model (e.g., Brown et al., 1994). The spatial discretization is based on the nodal DGM (e.g., Hesthaven & Warburton, 2008). To archive the eight-th order accuracy in terms of truncation errors, we use the hexahedral finite element with 83 nodes. As the numerical flux, we adopt the Rusanov flux for inviscid terms and the central flux for eddy viscous terms. As the spatial filter in LES, we use the16th-order elementwise modal filter with the exponential form. This filter rapidly removes the fluid structure whose wavelength is near two-grids. For this case, although the definition of spatial filter length seems to be not obvious, here we consider the filter length is essentially located between two- and four-grids.
2.2 Experimental setup
We conduct numerical experiments of idealized atmospheric boundary layer turbulence based on Nishizawa et al. (2015). The domain size is 9.6x9.6x3 km3. Initially, the horizontal wind is 5m/s, and the perturbation of potential temperature is added into stably stratified atmosphere. The constant heat flux with 200 W/m2 is injected at the surface. We perform the integration over four hours. The size of finite element is set to 200 m, which means the corresponding grid size in FDM (Δx) is approximately 30 m. To obtain the knowledge of the dynamic-physics coupling in DGM, we investigate how sensitive is the intensity of the modal filter (represented by the decay time for maximum wavenumber, Tnd) and the spatial filter length for turbulent model (Δsgs) to the energy spectra. We set these parameters such that Tnd/Δt~0.1, 10, 100 (where Δt is the time step), and Δsgs/Δx=1, 2, 4. Here, we regard the case of Tnd/Δt~10 and Δsgs/Δx=2 as the control experiment.
2.3 Results
We show the energy spectra of three-dimensional winds in the figure. For comparison, the results of SCALE-RM using the conservative FDM are also shown by gray lines. The slope for the control experiment becomes steeper than that of the -5/3 power law at wavelengths shorter than 200 m. Over the wavelength between 500 m and 200 m, the spectra for the control experiment and the stronger modal filter case has a little shallower slope than that of the -5/3 power law. This indicates the accumulation of energy. The sensitivity of the spatial filter length is higher than that of the modal filter’s intensity. In terms of the dynamics-physics coupling considering the effective resolution of dynamics, we expected that Δsgs/Δx= 4 would be appropriate in this experiment, however the actual spectra for Δsgs/Δx= 4 show too strong dissipation and deviate significantly from the -5/3 power law. To consider the reasons for the above results, we will revisit the formulation of LES in DGM. We will also conduct additional experiments with the higher resolution to make precise discussion, because the resolved inertial subrange is slightly narrow.