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  • View in gallery

    A schematic of a cubed sphere displayed with the global relief data. Each face (thick red solid lines) of the cubed sphere is divided into 4 × 4 elements (thick black solid lines), and each element has 4 × 4 GLL grid points (thin black solid lines).

  • View in gallery

    Speedup of the parallel computation for the dynamical core (a solid line with circles) on Tachyon II supercomputer system of KISTI, where the grid resolution is given as Ne = 60 and Np = 4, and the number of vertical layers is 30. The maximum number of processors used for the computation is 5 400. The dashed line shows perfect parallel scalability.

  • View in gallery

    Zonal wind for the baroclinic instability test (η = 0.69) at day 9 produced by the dynamical core using (a) 4th-order global filter, (b) 8th-order global filter, (c) 4th-order local-domain filter, (d) 8th-order local-domain filter, (e) 4th-order explicit diffusion, and (f) the spherical harmonics spectral model with an 8th-order filter. The contour interval is 4 m s−1. Numerals in the upper right of each panel indicate minimum and maximum values of the zonal wind field.

  • View in gallery

    As in Fig. 3, but for the sea level pressure field. The contour interval is 10 hPa.

  • View in gallery

    As in Fig. 3, but for day 30. The contour interval is 6 m s−1, and the thick lines indicate zero-contour line.

  • View in gallery

    Kinetic energy spectra (η = 0.69) at day 30 produced by the dynamical core using the 4th-order global filter (blue solid), the 8th-order global filter (blue dots), the 4th-order local-domain filter (red solid), the 8th-order local-domain filter (red dots), the spherical harmonics spectral model with the 8th-order spectral filter (gray solid), and the 4th-order explicit diffusion (black solid).

  • View in gallery

    Kinetic energy spectra (η = 0.69) for the high-resolution (about 0.25° at the equator) simulation of (a) the SEDC with a 4th-order filter, (b) the SEDC with an 8th-order filter, (c) the SEDC with a 4th-order explicit diffusion, and (d) the SHDC with an 8th-order spectral filter. From initial time to day 27 with a 3-day interval, colors of the solid lines of the kinetic energy spectra gradually change from light gray to black. The red solid line indicates the kinetic energy spectra at day 30.

  • View in gallery

    The Schär-type surface topography used in the experiments of the quiescent atmosphere in the presence of orography for (a) the CTL, (b) the H20, and (c) the O05 experiments. The embedded plot at the lower left of each panel is the surface height (m) along the equator.

  • View in gallery

    The time series of the maximum wind speeds using (left) the explicit diffusion and (right) the local-domain high-order filter. Symbols ν2.0x and ν0.5x indicate experiments with a double and halved diffusion coefficient, respectively.

  • View in gallery

    The horizontal distribution of the wind speed (m s−1) at the lowest model level by day 6 using the (left) explicit diffusion, (center) 4th-order global filter, and (right) 4th-order local-domain filter for the (top) CTL, (middle) H20, and (bottom) O05 test cases, where EXP, GIMP, and LIMP represent the explicit diffusion, the global high-order filter, and the local-domain high-order filter, respectively.

  • View in gallery

    The radius–η cross section of the wind speed (m s−1), azimuthally averaged over the center of topography, using the (left) explicit diffusion and (right) 4th-order local-domain filter for the (top) CTL, (middle) H20, and (bottom) O05 test cases, where EXP, GIMP, and LIMP represent the explicit diffusion, the global high-order filter, and the local-domain high-order filter, respectively.

  • View in gallery

    The vertically averaged kinetic energy spectra at day 6 using the local-domain high-order filter (solid line) and the explicit diffusion (dashed line) for the (a) CTL, (b) H20, and (c) O05 experiments, where EXP and LIMP represent the explicit diffusion and the local-domain high-order filter, respectively.

  • View in gallery

    Wall-clock time histogram for 6 days of time integration using 25 processors in parallel computing of the dynamical cores adopting the explicit diffusion (EXP), the 4th-order local-domain filter (LIMP), and the 4th-order global filter (GIMP). Numerals in the histogram indicate the wall-clock time in seconds.

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Effect of a High-Order Filter on a Cubed-Sphere Spectral Element Dynamical Core

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  • 1 Department of Environmental Atmospheric Sciences, Pukyong National University, and BK21 Plus Project of the Graduate School of Earth Environmental Hazard System, Busan, South Korea
  • | 2 Department of Environmental Atmospheric Sciences, Pukyong National University, Busan, South Korea
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Abstract

A high-order filter for a cubed-sphere spectral element model was implemented in a three-dimensional spectral element dry hydrostatic dynamical core. The dynamical core incorporated hybrid sigma–pressure vertical coordinates and a third-order Runge–Kutta time-differencing method. The global high-order filter and the local-domain high-order filter, requiring numerical operation with a huge sparse global matrix and a locally assembled matrix, respectively, were applied to the prognostic variables, except for surface pressure, at every time step. Performance of the high-order filter was evaluated using the baroclinic instability test and quiescent atmosphere with underlying topography test presented by the Dynamical Core Model Intercomparison Project. It was revealed that both the global and local-domain high-order filters could better control the numerical noise in the noisy circumstances than the explicit diffusion, which is widely used for the spectral element dynamical core. Furthermore, by adopting the high-order filter, the effective resolution of the dynamical core could be increased, without weakening the stability of the dynamical core. Computational efficiency of the high-order filter was demonstrated in terms of both the time step size and the wall-clock time. Because of the nature of an implicit diffusion, the dynamical core employing this filter can take a larger time step size, compared to that using the explicit diffusion. The local-domain high-order filter was computationally more efficient than the global high-order filter, but less efficient than the explicit diffusion.

Denotes content that is immediately available upon publication as open access.

This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/licenses/by/4.0/).

© 2018 American Meteorological Society.

Corresponding author: Hyeong-Bin Cheong, hbcheong@pknu.ac.kr

Abstract

A high-order filter for a cubed-sphere spectral element model was implemented in a three-dimensional spectral element dry hydrostatic dynamical core. The dynamical core incorporated hybrid sigma–pressure vertical coordinates and a third-order Runge–Kutta time-differencing method. The global high-order filter and the local-domain high-order filter, requiring numerical operation with a huge sparse global matrix and a locally assembled matrix, respectively, were applied to the prognostic variables, except for surface pressure, at every time step. Performance of the high-order filter was evaluated using the baroclinic instability test and quiescent atmosphere with underlying topography test presented by the Dynamical Core Model Intercomparison Project. It was revealed that both the global and local-domain high-order filters could better control the numerical noise in the noisy circumstances than the explicit diffusion, which is widely used for the spectral element dynamical core. Furthermore, by adopting the high-order filter, the effective resolution of the dynamical core could be increased, without weakening the stability of the dynamical core. Computational efficiency of the high-order filter was demonstrated in terms of both the time step size and the wall-clock time. Because of the nature of an implicit diffusion, the dynamical core employing this filter can take a larger time step size, compared to that using the explicit diffusion. The local-domain high-order filter was computationally more efficient than the global high-order filter, but less efficient than the explicit diffusion.

Denotes content that is immediately available upon publication as open access.

This article is licensed under a Creative Commons Attribution 4.0 license (http://creativecommons.org/licenses/by/4.0/).

© 2018 American Meteorological Society.

Corresponding author: Hyeong-Bin Cheong, hbcheong@pknu.ac.kr

1. Introduction

Numerical weather prediction (NWP) models usually incorporate horizontal diffusion or a filter scheme to control the grid-scale noise, which cannot be resolved for a given horizontal resolution, as well as to mimic the horizontal mixing (Boyd 1996; Gelb and Gleeson 2001; Majewski et al. 2002; Tomita and Satoh 2004; Knievel et al. 2007; Ullrich and Norman 2014). When the nonlinear model is time marched, the energy cascade, from large to small scale, is generated and leads to the accumulation of energy and enstrophy at the grid scale in the models. These unresolved scale disturbances tend to be aliased to the resolved scale during marching-in-time and will finally contaminate the numerical solution of the model. The horizontal diffusion process in the numerical model acts as the sink of energy and enstrophy so that it can prevent the buildup of spurious energy and enstrophy (Smagorinsky 1963; Shapiro 1970; Leith 1971; Xue 2000; Skamarock 2004; Cheong et. al 2004; Jablonowski and Williamson 2011; Marras et al. 2015).

When numerical noise is intended to be suppressed by diffusion schemes, it is required that well-resolved scales are diffused as little as possible. As discussed by several studies (MacVean 1983; Gelb and Gleeson 2001; Skamarock 2004; Jablonowski and Williamson 2011), second-order diffusion schemes are not scale selective because physically meaningful scales are dampened too much. Fourth-order diffusion schemes, which are more scale selective, have been most widely used in NWP models, and higher-order (6th- and 8th-order) diffusion schemes have also been adopted in some NWP models. The high-order diffusion scheme guarantees not only model stability, but also robustness at the model-resolvable scale. The effective resolution, defined as the decaying of the tail of the model kinetic energy spectra near the grid scale (Skamarock 2004), can also be increased by the high-order diffusion scheme. Therefore, the grid-scale-resolving performance of the global models is eventually dependent on the diffusion or filter schemes.

The numerical diffusion is implemented in several ways, depending on the numerical methods used to discretize partial differential equations. In the conventional spectral method using spherical harmonics functions (SHFs), the implicit horizontal diffusion (or high-order spectral filter), which is free of Courant–Friedrichs–Lewy (CFL) restrictions, is carried out in spectral space with ease by taking advantage of the fact that the eigenfunctions of the Laplace equation in the spherical coordinate system are SHFs (Sardeshmukh and Hoskins 1984; Collins et al. 2004; Takahashi et al. 2006). The double Fourier series spectral method (Cheong 2000), employing finite Fourier series both in the longitude and latitude, deals with the only tridiagonal matrix used for the implicit hyperdiffusion by successive inversion of the multiple Helmholtz equation matrices comprising the high-order filter matrix (Cheong et al. 2004). The gridpoint-based global models, incorporating horizontal discretization methods such as the finite difference method (FDM), the finite volume method, and the spectral element method (SEM) with nodal basis, usually adopt the explicit diffusion by adding an artificial term in the governing equation. Among a variety of the gridpoint models, the global models based on the cubed-sphere SEM are now widely accepted in the global modeling community because of their accuracy, grid flexibility, and parallel scalability (Giraldo and Rosmond 2004; Wang et al. 2007; Dennis et al. 2012; Evans et al. 2013; Choi and Hong 2016). A favorable choice of the order of diffusion scheme for the model is usually limited to the 4th-order diffusion (Dennis et al. 2012) because of the CFL restriction on the determination of a proper viscosity coefficient (Jablonowski and Williamson 2011; Cheong and Kang 2015), which becomes more serious as the order of the diffusion increases. Although the restriction can be avoided by the implicit treatment of the diffusion scheme, the implicit diffusion requiring inversion of the Helmholtz equation–type filter matrix can adversely affect the computational efficiency as well as the scalability.

Recently, Kang and Cheong (2017, hereafter KC17) developed a computationally efficient and parallel scalable high-order spatial filter for the spectral element model on the cubed sphere, where the filter equation is the high-order Helmholtz equation that corresponds to the implicit time differencing of a diffusion equation employing the high-order spherical Laplacian. Two types of the high-order filter, the global and local-domain high-order filters, were developed using the global and local-domain high-order Helmholtz equation matrices, respectively. As shown in KC17, the local-domain high-order filter has better computational efficiency than the global high-order filter without significant degradation of accuracy since it deals with the local filter matrix, consisting of 5 × 5 elements on the cubed sphere. Such a computational efficiency of local-domain filter over the global-domain filter was also demonstrated by Park et al. (2011) using the Fourier spectral method. By adopting the matrix operation method, which makes filtering a matrix–vector multiplication process, two high-order filters achieve near-perfect parallel scalability. However, the effect of high-order filters on the time-dependent flow was verified only in a two-dimensional shallow water equation model.

In the present study, the high-order filter for the cubed-sphere spectral element model developed by KC17 was implemented in a three-dimensional spectral element dry hydrostatic dynamical core (SEDC), and the performance of the high-order filter was evaluated by comparing it with the explicit diffusion, which is widely used for the spectral element dynamical core on the cubed sphere. The cubed-sphere SEDC is introduced in the following section, and the implementation of the high-order filter in the dynamical core is described briefly in section 3. The performance and computational efficiency of the high-order filter are evaluated in section 4. The present study is summarized in section 5.

2. The cubed-sphere spectral element dry hydrostatic dynamical core

The cubed-sphere SEDC, which is characterized by an accurate, conservative, and highly scalable dynamical core, was used in the present study. The dynamical core incorporates the SEM on the cubed sphere as a horizontal discretization method, a hybrid terrain–following sigma coordinate (Simmons and Burridge 1981) with a vertical FDM, and the strong stability-preserving Runge–Kutta third-order scheme (SSP-RK3; Gottlieb 2005) for the temporal discretization. The vector-invariant form of the dry hydrostatic governing equations, consisting of the momentum equation, thermodynamic equation, continuity equation, and hydrostatic equation, is introduced as follows (Zhang and Rančić 2007; Dennis et al. 2012):
e1a
e1b
e1c
e1d
e1e
where xi(i = 1, 2) is the local coordinate defined on each face, ui(i = 1, 2) and ui(i = 1, 2) are the covariant vector and the contravariant vectors, respectively, T is temperature, G = det(Gij) indicates the determinant of the metric tensor, ν denotes the diffusion coefficient, Φ represents the geopotential, and f refers to the Coriolis parameter. The κ equals Rd/cp, where Rd and cp are the ideal gas constant and the specific heat at constant pressure, respectively. The energy term E, vorticity ζ, and divergence D are written as
e2a
e2b
e2c
Vectors between covariant and contravariant components are converted by a metric tensor:
e3a
e3b
where . The transformation of the wind vector between the sphere and cube coordinates can also be done by using the transformation matrix of equiangular central projection [see section 4 in Nair et al. (2005)]:
e4a
e4b
The pressure is diagnosed by the hybrid coefficients A(η) and B(η), which are designed to follow the pressure and sigma coordinates, respectively, as defined by
e5
where p is pressure and ps denotes surface pressure. The quantity B(η) should be unity at the lowest model level (η = 1), and depth of the model atmosphere is determined by A(η) at the highest model level (η = ηtop). The boundary condition of the vertical velocity in the hybrid vertical coordinate is as follows:
e6
Defining the hybrid coefficients is somewhat arbitrary. In the present study, we used the hybrid coefficients presented by the Dynamical Core Model Intercomparison Project (DCMIP; Ullrich et al. 2012).
The SEM that calculates a physical quantity expanded by the polynomial basis for each element is used as a horizontal discretization method (Patera 1984; Taylor et al. 1997). The entire computational domain is divided into a finite number of nonoverlapped elements. The information at the boundaries of each element is shared to make the global continuous spectral element solution so that C0 continuity is enforced. The SEM has both the advantages of the grid flexibility of the finite-element method and the high-order accuracy of the spectral method. The polynomial expansion for a two-dimensional arbitrary variable U on each element is written as
e7
where N is the order of the polynomial basis function L. The Gauss–Lobatto–Lagrange-interpolating polynomials (GLLIPs) were employed as a basis of the polynomial defined on the Gauss–Lobatto–Legendre (GLL) grid (Rønquist 1988):
e8
where PN is the Legendre polynomial of order N defined on the GLL grid points xl [l = 0, 1, …, N], and k means the kth polynomial basis. The six faces of the cube are divided into 6(Ne × Ne) elements, and an element has Np × Np GLL grid points, where Np = N + 1. An example of a cubed-sphere grid (Ne = 4, Np = 4) is illustrated in Fig. 1.
Fig. 1.
Fig. 1.

A schematic of a cubed sphere displayed with the global relief data. Each face (thick red solid lines) of the cubed sphere is divided into 4 × 4 elements (thick black solid lines), and each element has 4 × 4 GLL grid points (thin black solid lines).

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

To diminish the numerical instability due to the oscillatory nature of the SEM operators during time integration of the system of the primitive equations (Taylor et al. 2009), a diffusion process should be implemented in the dynamical core. The biharmonic-type (∇4) explicit hyperviscosity operator is widely used, as in other dynamical cores incorporating the SEM, such as the spectral element version of the Community Atmosphere Model (CAM-SE) (Neale et al. 2010; Dennis et al. 2012) and Nonhydrostatic Unified Model of the Atmosphere (NUMA; Giraldo et al. 2013). In the present study, we adopted the explicit diffusion used in the CAM-SE. This hyperviscosity operator incorporating auxiliary vectors is solved explicitly using a mixed finite element integrated-by-parts formulation following Giraldo (1999). The hyperviscosity operator is applied to the prognostic variable, except for surface pressure, for each element after the last stage of SSP-RK3. The performance of the explicit diffusion is compared to the implicit diffusion, which will be presented in the next section.

It is well known that the cubed-sphere spectral element model has excellent parallel scalability. In the CAM-SE, the space-filling curve algorithm is employed for parallelization (Dennis et al. 2012). In the present study, parallelization of the hydrostatic dynamical core is very similar to that of the high-order filter (KC17), except for the exchange of the data at the boundaries of each element. The hydrostatic dynamical core constructed in this study was tested on the Tachyon II supercomputer system of the Korean Institute of Science and Technology Information (KISTI), having thousands of cores. The SEDC has the near-perfect scalability up to 5400 cores for a 0.5° global resolution (Ne = 60, Np = 4) with 30 vertical layers (Fig. 2), which is comparable to the parallel scalability of other dynamical cores incorporating the SEM, such as CAM-SE (Evans et al. 2013) and NUMA (Müller et al. 2018).

Fig. 2.
Fig. 2.

Speedup of the parallel computation for the dynamical core (a solid line with circles) on Tachyon II supercomputer system of KISTI, where the grid resolution is given as Ne = 60 and Np = 4, and the number of vertical layers is 30. The maximum number of processors used for the computation is 5 400. The dashed line shows perfect parallel scalability.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

3. Implementation of the high-order filter

The global and local-domain high-order filters, which are implicit-type filters, were developed by KC17. In this section, we briefly explain the procedure for constructing and implementing of the high-order filter.

The high-order filter equation (Cheong et al. 2004) can be obtained from the high-order implicit diffusion equation for an arbitrary variable Q:
e9a
e9b
where the former (the latter) is an explicit (implicit) filter, q being a positive is the order of the filter, and and are a filtered variable and a variable to be filtered, respectively. The high-order filter equation is horizontally discretized for each element using the SEM. For example, the matrix equation of Eq. (9b), the implicit filter, is written as
e10
where is the identity matrix, is a Laplacian operator matrix whose size is (), q is grid data of an element, and is a discrete filter matrix for each element. The filter matrix for all of the elements on the cubed sphere is assembled to represent the global high-order filter matrix, whereas only 5 × 5 elements are connected to represent the local-domain high-order filter that is introduced to reduce the computational cost of the global high-order filter. The filter matrix is inverted in advance, only once before time integration of the SEDC, so that matrix–vector multiplication can achieve the filtering process. As shown in KC17, the computational time for the inversion of the local high-order filter matrix is significantly reduced, compared to that for the global high-order filter (see their Fig. 14). The detailed procedure for constructing the global and local-domain high-order filters can be found in KC17. One thing to be noted is that the local solver (the local filter) is actually a part of the global one. But if the magnitude of the filter coefficient is very large, the accuracy of the local solver would not match the global one. Some data-analysis procedures, such as scale decomposition, require a large filter coefficient. In this case, utilization of the global filter would be more adequate. On the other hand, the filter as a numerical stabilizer during time integration requires a small value of the filter coefficient, since grid-size-scale noise is a main target scale. Therefore, the local filter used as a diffusion scheme for the dynamical core can take advantages of both computational efficiency and accuracy (or asymptotic convergence).

The implementation of the global and local-domain high-order filters in the three-dimensional dynamical core is very similar to that in the shallow water model (KC17), except for the application of the horizontal filter for each vertical layer. In the dry hydrostatic dynamical core, the high-order filter was applied to the prognostic variables, such as u, υ, and T, except for surface pressure at every time step after the SSP-RK3 time-stepping procedure. Parallelization of the high-order filters is also described in KC17 (in their section 4), and the parallel scalability of the global and local-domain high-order filters was almost ideal, scaling up to 1350 cores.

When the Laplacian operator is used as the numerical diffusion scheme and applied to the prognostic variables, careful choice of the vector or scalar Laplacian operator is required. The prognostic variables in the vector-invariant form of the governing equation incorporated in this study consist of u, υ, T, and ps. The high-order filter, based on the vector Laplacian operator (2), should be used to filter out the wind vector u and υ. The high-order filter used in this study, however, was constructed using the scalar Laplacian operator. By means of the definition of the vector Laplacian, we can apply the scalar Laplacian operator to the wind vector field as follows:
e11
where Vx, Vy, and Vz are the components of V. The wind vector, which is the function of λ and θ on each cubed-sphere grid point, can be transformed into components in the Cartesian coordinate (x, y, z) by using the following relations:
e12
where u and υ are the zonal and meridional winds, respectively. The high-order filter can then be applied to each component of V. After filtering the vector components, they should be transformed again to the wind vector on the cubed sphere by using the inverse relation of Eq. (12):
e13
where the asterisk denotes the filtered variable. The scalar variable, such as temperature T, is simply smoothed out by direct application of the high-order filter.

The viscosity coefficient of the high-order filter can be defined theoretically using the eigenvalue of the Laplacian operator for the cubed-sphere SEM (Cheong and Kang 2015; KC17), as in spectral filters for the spectral transform models (MacVean 1983; Roeckner et al. 2003; Jablonowski and Williamson 2011). However, in the present study, we used the diffusion coefficient for the explicit diffusion in CAM-SE (Dennis et al. 2012) to see the effect of the different diffusion schemes having the same diffusion coefficient.

4. Performance evaluation of the high-order filter

In this section, the performance of the high-order filter is evaluated through test cases for the dynamical core presented by the DCMIP (Ullrich et al. 2012). The baroclinic instability test (referred to as test case 4–1-0 in DCMIP) and the quiescent atmosphere in the presence of surface topography test (test case 2–0-0 in DCMIP) were chosen. The model configurations for the test cases are given in Table 1.

Table 1.

Model configuration of the dynamical core for each test case chosen in this study. Test cases 4–1-0 and 2–0-0, which are named in the DCMIP documents, indicate the baroclinic instability test and the quiescent atmosphere in the presence of the surface topography test, respectively. Values in parentheses denote those values used for the higher-resolution simulation.

Table 1.

a. Baroclinic instability test

For this test, a small Gaussian-type perturbation of zonal wind was added to the balanced steady-state model atmosphere. The perturbation growth in the form of baroclinic waves occurred during 15 days from the time of integration, after which the waves broke, merged, and dissipated. As recommended in the DCMIP test case document (Ullrich et al. 2012), the dynamical core was run for 30 days. The spherical harmonics hydrostatic dynamical core (SHDC; Simmons and Burridge 1981) with T120 resolution, employing the 8th-order spectral filter and semi-implicit time stepping, was also used as a reference model for the comparison. In this dynamical core, the prognostic variables are filtered out in spherical harmonics spectral space.

Figure 3 and 4 show the zonal wind field (η = 0.69) and surface pressure field at day 9, respectively. The zonal wind and surface pressure fields of the six experiments exhibited very similar distribution even though different diffusion schemes and dynamical cores were used, since perturbations grew linearly at this time, as in other studies (Jablonowski and Williamson 2006; Cheong 2006; Lauritzen et al. 2010). It appears that the difference between different orders of diffusion schemes (e.g., 4th- and 8th-order local-domain filter for the SEDC) is larger than the difference between different dynamics (e.g., the SEDC and SHDC). As can be seen in the right panels of Fig. 3, slightly larger amplitudes in the zonal wind fields were observed in the results for which the higher-order (i.e., 8th order) diffusion schemes were used. After day 9, the baroclinic wave continued to evolve with time until day 15 (not shown), after which it was broken and mixed nonlinearly until the end of the simulation. As shown in Fig. 5, the spatial distributions of the zonal wind fields at day 30 are very complicated. It is hard to find common features among the results, but it is evident that the 8th-order diffusion leaves more small-scale perturbations than the 4th-order diffusion.

Fig. 3.
Fig. 3.

Zonal wind for the baroclinic instability test (η = 0.69) at day 9 produced by the dynamical core using (a) 4th-order global filter, (b) 8th-order global filter, (c) 4th-order local-domain filter, (d) 8th-order local-domain filter, (e) 4th-order explicit diffusion, and (f) the spherical harmonics spectral model with an 8th-order filter. The contour interval is 4 m s−1. Numerals in the upper right of each panel indicate minimum and maximum values of the zonal wind field.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

Fig. 4.
Fig. 4.

As in Fig. 3, but for the sea level pressure field. The contour interval is 10 hPa.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

Fig. 5.
Fig. 5.

As in Fig. 3, but for day 30. The contour interval is 6 m s−1, and the thick lines indicate zero-contour line.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

To analyze the above features more quantitatively and gain an appreciation for the characteristics of the model diffusion, the kinetic energy spectrum at day 30 on that eta level was calculated. Because of the unstructured nature of the cubed-sphere grid system, the interpolation of the zonal and meridional wind fields from the cubed-sphere grid to the Gaussian latitude–longitude grid is required to calculate the kinetic energy spectra. We interpolated the horizontal wind fields using the spectral method, after which the fields were transformed into the spherical harmonics spectral space to compute the kinetic energy spectrum. Figure 6 shows the kinetic energy spectra (η = 0.69) at day 30. As expected, it is clearly seen that the kinetic energy spectra produced by the 8th-order diffusion possess more energy than those produced by the 4th-order diffusion, regardless of the different dynamics (e.g., the SEDC and the SHDC) and different filtering schemes (e.g., the explicit diffusion, the global and local-domain filters, and the spectral filter), and they have larger values at spherical wavenumbers larger than 40. Furthermore, the kinetic energy spectra of the 8th-order diffusion exhibit a slope slightly steeper than −3, whereas those of the 4th-order diffusion show a slope significantly steeper than −3. The kinetic energy of the 8th-order diffusion near higher wavenumbers is larger by almost an order of magnitude, compared with that of the 4th-order diffusion.

Fig. 6.
Fig. 6.

Kinetic energy spectra (η = 0.69) at day 30 produced by the dynamical core using the 4th-order global filter (blue solid), the 8th-order global filter (blue dots), the 4th-order local-domain filter (red solid), the 8th-order local-domain filter (red dots), the spherical harmonics spectral model with the 8th-order spectral filter (gray solid), and the 4th-order explicit diffusion (black solid).

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

For the simulations done with 1° horizontal resolution (as above), the dynamical cores incorporating the global and local-domain 8th-order filters and the spherical harmonics spectral filter produced kinetic energy spectra behaving as k−3, which is a canonical structure of the atmospheric synoptic-scale kinetic energy spectrum (Lindborg 1999; Skamarock et al. 2014). On the other hand, the dynamical cores using the 4th-order diffusion could not generate the canonical kinetic energy spectrum. Successful simulation of the model-derived canonical spectrum is dependent on the configuration of the model diffusion. This trend becomes more conspicuous as the horizontal resolution increases, since there is a transition between synoptic-scale and mesoscale regimes behaving approximately as k−3 to k−5/3 (Lindborg 1999; Skamarock 2004; Lovejoy et al. 2009; Waite and Snyder 2009; Skamarock et al. 2014). The spectral slope of −5/3 does not appear in the results of the simulation with 1° resolution since a grid scale of that moderate horizontal resolution still belongs to synoptic scale. To simulate mesoscale dynamics, we carried out the same test with higher resolution by configuring Ne = 120, Np = 4 for the SEDC and T480 for the SHDC, which are approximately equal to a 0.25° horizontal grid spacing at the equator. The time step size was set to 30 s and 90 s for the SEDC and the SHDC, respectively. In this test, we omitted the experiment with the global high-order filter for the SEDC because of its similarity to the local high-order filter.

Figure 7 shows the global kinetic energy spectra (η = 0.69) from the initial time to day 30 for the higher-resolution simulation. From initial time to day 9, the kinetic energy spectra of all experiments showed similar patterns because the baroclinic wave grew linearly during that period. After day 9, different characteristics of the spectral tail between 4th-order diffusion (Figs. 7a,c) and 8th-order diffusion (Figs. 7b,d) became clear at high wavenumbers larger than 100. The kinetic energy spectra at day 30 (a red solid line) using the 8th-order filter (Figs. 7b,d) reveal the transition from the synoptic scale to mesoscale, consistent with the high-resolution model-derived kinetic energy spectra in other studies (Hamilton et al. 2008; Terasaki et al. 2009; Evans et al. 2013; Skamarock et al. 2014). The transition of the kinetic energy spectra for the 8th-order diffusion simulation begins at approximately wavenumber 100, whereas the spectra of the 4th-order diffusion kept the slightly shallow slope of −3 (Figs. 7a,c). The spectral characteristics for the SEDC with the 4th-order diffusion are like those of Evans et al. (2013, their Fig. 6), although the simulation was different. The slope of the spectra of the 8th-order diffusion is slightly shallower than −5/3 after the transition and starts to decay at approximately wavenumber 300, where the kinetic energy spectra of the 8th-order diffusion are larger by almost an order of magnitude, compared to those of the 4th-order diffusion. It seems that the effective resolution of the SEDC, as defined by Skamarock (2004), increased by using the 8th-order filter. It is worth noting that the spectra of the 8th-order filter for different dynamics (i.e., the SEDC and SHDC) exhibited very similar slopes.

Fig. 7.
Fig. 7.

Kinetic energy spectra (η = 0.69) for the high-resolution (about 0.25° at the equator) simulation of (a) the SEDC with a 4th-order filter, (b) the SEDC with an 8th-order filter, (c) the SEDC with a 4th-order explicit diffusion, and (d) the SHDC with an 8th-order spectral filter. From initial time to day 27 with a 3-day interval, colors of the solid lines of the kinetic energy spectra gradually change from light gray to black. The red solid line indicates the kinetic energy spectra at day 30.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

b. Quiescent atmosphere in the presence of the surface topography

The numerical filter or diffusion scheme should successfully remove the numerical noise to guarantee a stable run of the dynamical core, even in a very noisy model atmosphere. The steady-state atmosphere at rest in the presence of the moderately steep orography test case (DCMIP test case 2–0-0) is a stricter test case than the baroclinic instability test in terms of the numerical instability of the dynamical core. As the complexity of the bottom topography increases, the numerical noise due to the pressure gradient term associated numerical error may increase. This numerical noise appears as the spurious flow during time integration.

For the control experiment (hereafter, CTL) the surface height was defined as (Ullrich et al. 2012):
e14
where h0(= 200 m) means the maximum mountain height, Rm(= 3π/4) is mountain radius, ζm(= π/16) is mountain oscillation half-width, and the great circle distance rm from the center of the mountain is written as
e15
where (λm, θm) is the center of the mountain. The surface topography for the CTL case is shown in Fig. 8a. The mountain appears to have small wavelength undulations whose amplitude decays gradually away from the center. To test the performance of the implicit viscosity in more noisy circumstances, the original, moderately steep topography was somewhat modified by changing the parameters in Eq. (14), such as the maximum height h0 and the scale of topography ζm. For the steeper surface topography, the maximum height of the mountain was doubled (Fig. 8b, hereafter H20). Also, more complex surface topography was obtained by halving the scale of undulation ζm (Fig. 8c, hereafter O05) The analytic viewpoint of this test case is that a motionless atmosphere is kept during time integration without any spurious flow. Thus, the spurious flow, which can be suppressed by the numerical filter or diffusion scheme, is regarded as the numerical error of the dynamical core. The steeper and more complex topography may lead to a violation of the CFL stability restriction of the explicit diffusion. This becomes more serious when the higher-order diffusion is applied. Therefore, we also compare the stability of the implicit and explicit diffusion schemes used for the SEDC in the next subsection. The model atmosphere is divided into 15 vertical hybrid levels, and the dynamical core was integrated for 6 days with a time step of 180 s. The horizontal resolution of the SEDC was set to 1° × 1°, which corresponds to Ne = 30 and Np = 4. The 4th-order diffusion was only used to focus on the effect of the implicit diffusion implemented in the SEDC.
Fig. 8.
Fig. 8.

The Schär-type surface topography used in the experiments of the quiescent atmosphere in the presence of orography for (a) the CTL, (b) the H20, and (c) the O05 experiments. The embedded plot at the lower left of each panel is the surface height (m) along the equator.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

Figure 9 shows the time series of the maximum wind speed (u2 + υ2)1/2 for each experiment. For all diffusion schemes, the H20 experiment has the largest numerical error, compared to other experiments. It seems that the SEDC had more difficulty in resolving the steep topography than the complex one. The maximum wind speed for all experiments of the explicit diffusion was about 2 times larger than that of the local-domain high-order filter. In particular, the H20 experiment using the explicit diffusion shows the largest error. It is worth noting that the time evolution of the H20 experiment using the explicit diffusion fluctuated with time, which means that the dynamical core incorporating the explicit diffusion was unstable. Contrary to this, the time series of the local-domain high-order filter was smooth and increased gradually with time. Regardless of the magnitude of the viscosity coefficient, furthermore, the local high-order filter always shows better performance than the explicit diffusion (lower panels of Fig. 9). Sensitivity of both filters to the viscosity coefficient is quite similar, but it seems that the local high-order filter can suppress the numerical noise better, especially at early stages of the simulation. For the smaller viscosity coefficient, both diffusion schemes are more sensitive than the larger viscosity coefficient, but error of the local high-order filter is still lower. The maximum wind speed using the global high-order filter is very similar to that of the local-domain high-order filter (not shown).

Fig. 9.
Fig. 9.

The time series of the maximum wind speeds using (left) the explicit diffusion and (right) the local-domain high-order filter. Symbols ν2.0x and ν0.5x indicate experiments with a double and halved diffusion coefficient, respectively.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

Shown in Fig. 10 are the horizontal distributions of the wind speed at the lowest level by day 6. Although the dynamical cores start with a resting atmosphere, the spurious flow was generated and spread over the mountain. Because of the shape of the surface topography, the circular pattern of the wind speed field was dominant. High wind speed, especially, was generated commonly in all experiments around the center of the topography. The strongest wind is observed in the H20 experiment, whereas the spurious flow is most widely distributed for more complex topography (O05 experiment). It is worth noting that the results of the global and local-domain high-order filters are almost the same, even though different types of bottom topography were used for the experiments. As compared to the two high-order filters (i.e., the global and local-domain high-order filter), the wind speed using the explicit diffusion shows larger magnitudes (i.e., larger numerical error) and a greater spread in pattern, especially for the H20 case, which is the experiment with the steeper mountain. To see the vertical distribution of the wind speed, we determined the azimuthal average of the wind speed field above the center of the mountain (λm, θm) by noticing that the wind speed field has a circular pattern. The radius–eta cross section of the azimuthally averaged wind speed is shown in Fig. 11. For both the explicit diffusion and the local-domain high-order filter, the spurious wind is vertically spread over the entire model layer, and it is strong near the center of the mountain, as expected from the horizontal distribution. The flow oscillated horizontally but stretched vertically because the error source stems from the wavy form of the surface topography. For the local-domain high-order filter, a more stretched flow near the top of the model was observed around the center of the mountain. However, in the middle and lower atmospheres, the spurious flow using the local-domain high-order filter was significantly alleviated, particularly in the radius from about 10° to about 60°, regardless of the shape of the surface topography.

Fig. 10.
Fig. 10.

The horizontal distribution of the wind speed (m s−1) at the lowest model level by day 6 using the (left) explicit diffusion, (center) 4th-order global filter, and (right) 4th-order local-domain filter for the (top) CTL, (middle) H20, and (bottom) O05 test cases, where EXP, GIMP, and LIMP represent the explicit diffusion, the global high-order filter, and the local-domain high-order filter, respectively.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

Fig. 11.
Fig. 11.

The radius–η cross section of the wind speed (m s−1), azimuthally averaged over the center of topography, using the (left) explicit diffusion and (right) 4th-order local-domain filter for the (top) CTL, (middle) H20, and (bottom) O05 test cases, where EXP, GIMP, and LIMP represent the explicit diffusion, the global high-order filter, and the local-domain high-order filter, respectively.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

The dynamical core incorporating the implicit diffusion, as shown, had fewer numerical errors than that using the explicit diffusion in the noisy circumstance. This may result from the efficient removal of the small-scale numerical noise by the implicit diffusion. To see this feature quantitatively, we display the vertically averaged kinetic energy spectrum by day 6 in Fig. 12. As it should be, it is clearly seen that a decrease in kinetic energy across all experiments is observed for the local-domain high-order filter. Most of all, the kinetic energy spectrum at wavenumbers approximately larger than 60, for the local-domain high-order filter, is much weaker than that for the explicit diffusion. Thus, the high-order filter diminishes the small-scale numerical noise more efficiently than the explicit diffusion, and eventually, weaker spurious flow (or less numerical noise) was generated by the surface topography. The largest differences in the kinetic energy spectra between the two different diffusion schemes (i.e., the local-domain high-order filter and the explicit diffusion) were noted for the H20 experiments, while the O05 experiments showed the smallest differences. It might be thought that the high-order filter can contribute to the stability of the SEDC by reducing the numerical errors arising from steep gradient of topography.

Fig. 12.
Fig. 12.

The vertically averaged kinetic energy spectra at day 6 using the local-domain high-order filter (solid line) and the explicit diffusion (dashed line) for the (a) CTL, (b) H20, and (c) O05 experiments, where EXP and LIMP represent the explicit diffusion and the local-domain high-order filter, respectively.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

c. Effect of the high-order filter on the time step size

The high-order filters used in this study are implicit-type filters that are free of the CFL restriction. Furthermore, especially for the cubed-sphere SEM, the high-order filters can prevent the numerical noise from the C0 enforced element boundaries by taking advantage of the filter weights covering several elements (KC17). Therefore, the spectral element dynamical core with the high-order filter can take larger time step size than that with the explicit diffusion. The same test case used in the previous section was chosen to test the limitation of the time step size. The H20 experiment was carried out to demonstrate more clearly the difference between the implicit and explicit diffusion schemes. The model configurations were also the same, except for the time step size and the filter order. The results are shown in Table 2, where the crisscross notation “×” means that the dynamical core blew up before day 6. It is evident that the explicit diffusion is much more vulnerable to the numerical instability than the implicit diffusion. It was possible to set the time step size of the 6th-order global and local-domain filters to 3 times larger than that of the explicit diffusion. Although the dynamical core employing the 4th-order explicit diffusion was successfully terminated by day 6, the numerical error was larger than that for the high-order filters. As the diffusion order increases, the time step size of the explicit diffusion is reduced, while that of the high-order filters is maintained. It is worth noting that the numerical errors of the high-order filters for different time step sizes were very similar to one another for the same order of diffusion, while that of the 4th-order explicit diffusion was largely dependent on the time step size.

Table 2.

The maximum wind speed (m s−1) at day 6 for the simulation of the steady-state atmosphere at rest with surface topography, where EXP, GIMP, and LIMP represent the explicit diffusion, the global high-order filter, and the local-domain high-order filter, respectively. Numerals in the first row represent the time step size in seconds, and the symbol × denotes the model blow-up before day 6.

Table 2.

d. Computational efficiency of the local-domain high-order filter

The local-domain high-order filter, which incorporates multiple cells, was developed to reduce the computational cost of the global high-order filter that requires inversion of the huge global filter matrix and the huge sparse matrix–vector multiplication for the filtering. As shown in the previous sections, it was found that performance of the local-domain high-order filter was comparable to that of the global high-order filter. Thus, the local-domain high-order filter can take advantage of the computational efficiency without degrading the performance. Using the same test as in section 4b, the computational efficiency of the local-domain high-order filter was investigated.

Figure 13 shows the wall-clock time of the SEDC with three types of 4th-order diffusion schemes run for 6 days of time integration using 25 cores. The SEDC with the global high-order filter, as expected, required the longest time to complete the simulation. The wall-clock time of the local-domain high-order filter was reduced by a factor of 1.5, compared to the global high-order filter, while it was larger than that of the explicit diffusion by a factor of 1.6.

Fig. 13.
Fig. 13.

Wall-clock time histogram for 6 days of time integration using 25 processors in parallel computing of the dynamical cores adopting the explicit diffusion (EXP), the 4th-order local-domain filter (LIMP), and the 4th-order global filter (GIMP). Numerals in the histogram indicate the wall-clock time in seconds.

Citation: Monthly Weather Review 146, 7; 10.1175/MWR-D-17-0226.1

5. Summary

The global and local-domain high-order filters developed by KC17 were implemented in the three-dimensional spectral element dry hydrostatic dynamical core on the cubed sphere, and the performance was evaluated using dynamical core test cases. The high-order filter was applied to the prognostic variables, except for surface pressure, after the RK3 time stepping. The filter matrix for both high-order filters was constructed only once before time integration and it was applied to the prognostic variables repeatedly at every time step. The parallel scalability of the dynamical core and high-order filter was near perfect, as in other spectral element dynamical cores, such as CAM-SE and NUMA.

Two kinds of dynamical core test results produced by DCMIP demonstrate the usefulness of the high-order filter implemented in the cubed-sphere hydrostatic dynamical core. In comparison to the explicit diffusion, which is widely used in the cubed-sphere spectral element model, the baroclinic instability test showed that the effective resolution increased by using the 8th-order filter, especially for the high-resolution simulation, maintaining the model stability simultaneously. The performance of the high-order filter was also tested using the steady-state atmosphere at rest with the surface orography. Over all experiments carried out in this test, the global and the local high-order filter can control the grid-scale numerical noise more efficiently than explicit diffusion in any cases, as revealed by vertically averaged kinetic energy spectra. Although the numerical procedure for the filtering was somewhat different, the global and local-domain high-order filters produced very similar results, as was expected.

Computational efficiency of the high-order filter was investigated in terms of the time step size and the wall-clock time of the dynamical core. It was found that the dynamical core using the high-order filter could take time step sizes several times larger than that using the explicit diffusion. The computational time of the dynamical core using the local-domain high-order filter decreased by a factor of 1.5, compared to the global high-order filter, while it increased only by a factor of 1.6, compared to the explicit diffusion. Evidently, the local-domain high-order filter is more computationally efficient without undergoing significant performance degradation, compared to the global high-order filter.

Acknowledgments

The authors thank anonymous reviewers for constructive comments. This research was supported by the Korea Institute of Atmospheric Prediction Systems in 2017. Hyun-Gyu Kang acknowledges the financial support of the postdoctoral fellow program through the BK21 Plus Project of the Graduate School of Earth Environmental Hazard System (Grant 21A20151713014).

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