The non-hydrostatic (NH) compressible Euler equations for dry atmosphere were solved in a simplified two-dimensional (2-D) slice framework employing a spectral element method (SEM) for the horizontal discretization and a finite difference method (FDM) for the vertical discretization. By using horizontal SEM, which decomposes the physical domain into smaller pieces with a small communication stencil, a high level of scalability can be achieved. By using vertical FDM, an easy method for coupling the dynamics and existing physics packages can be provided. The SEM uses high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL) quadrature points. The FDM employs a third-order upwind-biased scheme for the vertical flux terms and a centered finite difference scheme for the vertical derivative and integral terms. For temporal integration, a time-split, third-order Runge–Kutta (RK3) integration technique was applied. The Euler equations that were used here are in flux form based on the hydrostatic pressure vertical coordinate. The equations are the same as those used in the Weather Research and Forecasting (WRF) model, but a hybrid sigma–pressure vertical coordinate was implemented in this model.
We validated the model by conducting the widely used standard tests: linear hydrostatic mountain wave, tracer advection, and gravity wave over the Schär-type mountain, as well as density current, inertia–gravity wave, and rising thermal bubble. The results from these tests demonstrated that the model using the horizontal SEM and the vertical FDM is accurate and robust provided sufficient diffusion is applied. The results with various horizontal resolutions also showed convergence of second-order accuracy due to the accuracy of the time integration scheme and that of the vertical direction, although high-order basis functions were used in the horizontal. By using the 2-D slice model, we effectively showed that the combined spatial discretization method of the spectral element and finite difference methods in the horizontal and vertical directions, respectively, offers a viable method for development of an NH dynamical core.
There is growing interest in developing highly scalable dynamical cores using numerical algorithms under petascale computers with many cores (with the goal of exascale computing just around the corner), and the spectral element method (SEM), with high efficiency and accuracy, is known to be one of the most promising methods (Taylor et al., 1997; Giraldo, 2001; Thomas and Loft, 2002). SEM is local in nature because it has a large on-processor operation count (Kelly and Giraldo, 2012). SEM achieves this high level of scalability by decomposing the physical domain into smaller pieces with a small communication stencil. Additionally, SEM has been shown to be very attractive for achieving high-order accuracy and geometrical flexibility on the sphere (Taylor et al., 1997; Giraldo, 2001; Giraldo and Rosmond, 2004).
To date, SEM has been implemented successfully in atmospheric modeling, such as in the community atmosphere model–spectral element (CAM-SE) dynamical core (Thomas and Loft, 2005) and the scalable spectral element Eulerian atmospheric model (SEE-AM) (Giraldo and Rosmond, 2004). These models consider the primitive hydrostatic equations on global grids, such as a cubed sphere tiled with quadrilateral elements using SEM in the horizontal discretization and the finite difference method (FDM) in the vertical. The robustness of SEM has been illustrated through three-dimensional dry dynamical test cases (Giraldo and Rosmond, 2004; Giraldo, 2005; Thomas and Loft, 2005; Taylor et al., 2007; Lauritzen et al., 2010).
The ultimate objective of our study is to build a 3-D non-hydrostatic (NH) model based on the compressible Navier–Stokes equations using SEM in the horizontal discretization and FDM in the vertical. Because testing a 3-D NH model requires a large amount of computing resources, studying the feasibility of our approach in 2-D is an attractive alternative to the development of a fully 3-D model. This is the case because a 2-D slice model can effectively test the practical issues resulting from the vertical discretization and time integration prior to construction of a full 3-D model. Although we could discretize the vertical direction using SEM (as proposed in Kelly and Giraldo, 2012, and Giraldo et al., 2013), we chose to use a finite difference method for discretization in the vertical direction because it provides an easy way to couple the dynamics and existing physics packages.
For this objective, we developed a dry 2-D NH compressible Euler model based
on SEM along the
In this paper, we demonstrate the feasibility of the 2-D NH model by conducting conventional benchmark test cases and by focusing on the description of the numerical scheme for the spatial discretization. We verify the 2-D NH by analyzing six test cases: inertia–gravity wave, rising thermal bubble, density current wave, linear hydrostatic mountain wave, and tracer advection and gravity wave over the Schär-type mountain.
The organization of this paper is as follows. In the next section, we describe the governing equations with definitions of the prognostic and diagnostic variables used in our model. In Sect. 3, we explain the temporal and spatial discretization including the spectral element formulation. In Sect. 4, we present the results of the 2-D NH model using all four test cases, and finally, in Sect. 5, we summarize the paper and propose future directions.
We adopted the formulation of the governing-equation set of SK08. Here, we
implemented the hybrid sigma–pressure coordinate introduced in PK13, which only
considers the hydrostatic primitive equation. The hybrid sigma pressure
coordinate is defined with
The flux-form Euler equations for dry atmosphere to be recast using
perturbation variables are expressed as
For completeness, the diagnostic relation for
The above equation allows for
This concludes the description of the governing equations used in our model; in the next section, we describe the discretization of the continuous form of the governing equations that are used in our model.
For a given
The GLL points
We now introduce the polynomial expansions into our governing equations in
the form of
The right-hand sides of Eqs. (17) and (18) are evaluated using the Gaussian
quadrature of Eq. (15). It is noted that using GLL points for both
interpolation and integration results in a diagonal mass matrix
The horizontal derivatives included in the right-hand side of Eq. (17) are
evaluated using the analytic derivatives of the basis functions as follows:
Note that the non-differential operations, such as cross products, are computed directly at grid points since we use nodal basis functions associated with Lagrange polynomials based on the GLL points. In order to satisfy the equations globally, we use the direct stiffness summation (DSS) operation. For a more detailed description of the SEM, see Giraldo and Rosmond (2004), Giraldo and Restelli (2008), and Kelly and Giraldo (2012).
Using a Lorenz staggering, the variables
In addition to the governing equations, a viscous term might be needed to
conduct some tests. The viscosity used here is an explicit Laplacian
(
Grid points of columns within an element having four GLL points. The hybrid sigma–pressure coordinates are illustrated, and the closed (open) circles on the solid (dashed) line indicate the location of the variables at layer midpoints (interfaces).
To integrate the equations, we used the time-split RK3 integration technique following the strategy of SK08. In the time-split RK3 integration, low-frequency modes due to advective forcings are explicitly advanced using a large time step in the RK3 scheme, but high-frequency modes are integrated over smaller time steps. Among the high-frequency modes, horizontally propagating acoustic/gravity waves are advanced using an explicit forward–backward time integration scheme and vertically propagating acoustic waves and buoyancy oscillations are advanced using a fully implicit scheme (Klemp et al., 2007). For numeric stability, acoustic-mode filterings of the forward centering of the vertically implicit portion and divergence damping of the horizontal momentum equation are used, which is the same as in the WRF model (Skamarock et al., 2008). It is notable that the time-split RK3 integration scheme is third-order accurate for linear equations and second-order accurate for nonlinear equations (SK08).
This technique has been shown to work effectively within numerous
non-hydrostatic models, including the WRF model (Skamarock et al., 2008), the
Model for Prediction Across Scales (MPAS) (Skamarock et al., 2012), and the
Non-hydrostatic Icosahedral Atmospheric Model (NICAM) (Satoh et al., 2008).
It is also noted that, in the procedure of the time-split RK3 integration,
the difference between the approach used in this paper and that in SK08
comes from the vertical coordinate. Since we use the hybrid sigma–pressure
coordinate, the equation for
We validated the 2-D NH model with six test cases: linear hydrostatic mountain-wave, tracer-advection, and gravity-wave tests over Schär Mountain, as well as density current, inertia–gravity wave, and rising thermal bubble experiments. The last three cases do not have analytic solutions. Therefore, for the mountain experiments, the numerical results of the 2-D NH model were compared with analytic solutions (Durran and Klemp, 1983; Schär et al., 2002); for the other experiments, we compared our results with the results of other published papers.
It should be mentioned that the horizontal SEM formulation is able to
utilize arbitrary-order polynomials per element to represent the discrete
spatial operators, but in this paper all the results presented use either
fifth- or eighth-order polynomials. The averaged horizontal grid spacing is
defined as
Summary of the resolutions and time-step sizes used for the tests.
We simulated the linear hydrostatic mountain-wave test introduced by Durran and Klemp (1983) (DK83 hereafter) in which the analytic steady-state solution is provided by using a single-peak mountain with uniform zonal wind. To compare our results with the analytic and numerical solutions shown in DK83, the 2-D NH was initialized using the same initial conditions and mountain profile as in DK83, and we analyzed our results using the same metrics as DK83.
The mountain profile is given by
Steady-state flow of (top) horizontal velocity (m s
Vertical flux of horizontal momentum, normalized by its analytic
value at several nondimensional times
Figure 2 shows the numerical and analytic solutions at steady state for the
horizontal and vertical velocities, which agree reasonably well. The
vertical velocity fields match very closely, although the extrema in the
horizontal velocity field are underestimated by the numerical model. The
underestimated extrema in the horizontal velocity were also shown in both
models of DK83, which used
To check the vertical transport of horizontal momentum, Fig. 3 shows the
normalized momentum flux values at various times. It is observed that the
flux has developed well and that the simulations reach steady state after
Tracer advection test over the topography (red line).
Steady-state flow of
Potential temperature perturbation after 900 s using grid spacing of
In order to verify the feasibility of 2-D NH to treat steep surface elevations associated with the vertical terrain-following coordinate, we performed the tracer-advection and gravity-wave experiments introduced by Schär et al. (2002) (SC02 hereafter), in which the mountain is defined by a five-peak mountain chain, over the Schär-type mountain. To compare our results with the numerical solution shown in SC02, the initial conditions and mountain profiles are the same as those of SC02.
For the tracer-advection test, the mountain profile is given by
The numerical solutions and the error field are shown in Fig. 4. The figure
uses the same contouring interval as in SC02. Even at
The Schär-type mountain gravity-wave test was initialized in a stratified
atmosphere with the Brunt–Väisälä frequency of
Figure 5 shows the simulated results of the perturbed horizontal and vertical
wind speeds after 10 h. In comparison with the analytic solution, the
numerical solutions match quite well. The results of the present model are
also very similar to the results of other numerical models (Giraldo and
Restelli, 2008; Li et al., 2013). For a quantitative comparison, we present the
root-mean-square errors for
Root-mean-square errors (RMSEs) of the Schär-type mountain wave after 10 h
for
In order to verify the feasibility of 2-D NH to control oscillations with
numerical viscosity and evaluate numerical schemes in 2-D NH, we conducted
the density current test suggested by Straka et al. (1993). The density
current test is initialized using a cold bubble in a neutrally stratified
atmosphere. When the bubble touches the ground, the density current wave
starts to spread symmetrically in the horizontal direction, forming
Kelvin–Helmholtz rotors. Following Straka et al. (1993), we employed a
dynamic viscosity of
For an initial cold bubble, the potential temperature perturbation is given
as
Figure 6 shows the potential temperature perturbation after 900 s for 400,
200, 100, and 50 m grid spacings (
In order to examine the effect of higher order of the basis polynomials than
fifth-order, we show profiles of the potential temperature perturbation at the
height of 1200 m in the simulations using fifth-order polynomials together
with the simulations using eighth-order polynomials (Fig. 7). Note that the
simulations using eighth-order polynomials have the same number of GLL grid
points as the simulations using fifth-order basis polynomials. This was
achieved by using a lower number of elements in the eighth-order experiment than in
the fifth-order experiment as the number of grid points at a given level
becomes
Profiles of
Comparison between fifth- and eighth-order polynomials per element for
the density current. The simulation was conducted with a resolution of
In order to investigate the characteristics of the convergence more clearly,
a self-convergence test was carried out. For this test, a reference solution
is obtained by using spatial resolution
Self-convergence test for the density current test; relative L2
error norms of the potential temperature perturbation
This test examines the evolution of a potential temperature perturbation
Figure 9 shows the solution
Potential temperature perturbation at the initial time (top) and
time 3000 s (bottom) for
Profiles of potential temperature perturbation along the 5000 m
height for
Figure 10 shows profiles along 5000 m for various horizontal resolutions. All
models show consistently identical solutions with symmetric distribution
about the midpoint (
Comparison of the numerical results for various horizontal
resolutions for the inertia–gravity wave. All simulations use eighth-order
polynomials per element and a vertical resolution of
We also conducted the rising thermal bubble test to verify the consistency of the scheme in the model to simulate thermodynamic motion (Wicker and Skamarock, 1998). This test considers the time evolution of warm air in a constant potential temperature environment for an atmosphere at rest. The air that is warmer than ambient air rises due to buoyant forcing, which then deforms due to the shearing motion caused by gradients of the velocity field and eventually shapes the thermal bubble into a mushroom cloud. Because the test case has no analytic solution, the simulation results were evaluated qualitatively.
Plots of
The initial conditions we used follow those of GR08, in which the domain for
the case is defined as
Figure 11 shows the potential temperature perturbation, horizontal wind field,
and vertical wind field for the simulations of the two resolutions of 20
and 5 m horizontal and vertical grid spacings (
We also show the vertical profiles of potential perturbation at
Vertical profiles of the potential temperature perturbation for the
rising thermal bubble test at
Domain maximum potential temperature perturbation (top) and vertical wind (bottom) for the rising thermal bubble test. All simulations use fifth-order basis polynomials per element, and the vertical resolutions are the same as the horizontal resolutions.
The non-hydrostatic compressible Euler equations for a dry atmosphere were
solved in a simplified 2-D slice (
For the spatial discretization, the spatial operators were separated into their horizontal and vertical components. In the horizontal components, the operators were discretized using SEM, in which high-order representations are constructed through the GLL grid points by Lagrange interpolations in elements. Using GLL points for both interpolation and integration results in a diagonal mass matrix, which means that the inversion of the mass matrix is trivial. In the vertical components, the operators were discretized using the third-order upwind-biased finite difference scheme for the vertical fluxes and centered differences for the vertical derivatives. The time discretization relied on the time-split, third-order Runge–Kutta technique.
We presented results from idealized standard benchmark tests for large-scale flows (e.g., mountain-wave tests) and for non-hydrostatic-scale flows (e.g., inertia–gravity wave, rising thermal bubble, and density current). By varying the viscosity between test cases, the numerical results showed that the present dynamical core is able to produce high-quality solutions comparable to other published solutions. These tests effectively revealed that the combined spatial discretization method of the spectral element and finite difference methods in the horizontal and vertical directions, respectively, offers a viable method for the development of a NH dynamical core. Further work will be needed to achieve accurate solutions for a resting atmosphere over steep orography with minimal diffusion and to implement a horizontal diffusion operator in physical space, although horizontal diffusion on the coordinate surface was used in this study. Further research will also be conducted to couple the present core with the existing physics packages and extend the 2-D slice framework to develop a 3-D dynamical core for the global atmosphere in which the cubed-sphere grid is used for the spherical geometry.
This work was carried out through the R&D project on the development of global numerical weather prediction systems of the Korea Institute of Atmospheric Prediction Systems (KIAPS) funded by the Korea Meteorological Administration (KMA). The first author thanks Joseph B. Klemp for sharing his idea for the hybrid sigma–pressure coordinate and also thanks Francis X. Giraldo for his assistance and his MA4245 course at the Naval Postgraduate School, which introduced us to the spectral element method. The second author gratefully acknowledges the support of KIAPS, the Office of Naval Research through program element PE-0602435N, and the National Science Foundation (Division of Mathematical Sciences) through program element 121670. We also thank the reviewers for their constructive suggestions. Edited by: H. Weller