1. Introduction
Various methods have been developed to deal with the computational restrictions introduced by fast waves in the atmosphere. The first method arises from a modification of the equation set to “filter” out fast-moving waves. At large horizontal scales, hydrostatic models [including Taylor et al. (2008) and Lin (2004)] are usually employed to remove the acceleration of the vertical velocity from the system. This approach eliminates vertically propagating sound waves but reaches its limitations when the grid size is reduced to nonhydrostatic scales around 10 km. Motions at this scale can be dominated by large vertical velocities, and so the vertical acceleration term cannot be neglected. Further, the dispersion relation of the hydrostatic equation set reveals that the phase speed of gravity wave modes can be overestimated at these scales, as compared to the full nonhydrostatic equations (Durran 1999). Alternatively, so-called sound-proof systems of equations, including the Boussinesq equations, the anelastic system of Ogura and Phillips (1962), the pseudo-incompressible system of Durran (1989), and the unified approximation of Arakawa and Konor (2009), have been successfully used in models by removing sound waves from the governing equations. Nonetheless, it remains an open question as to whether these modified systems are valid on all scales [see, e.g., the discussion in Davies et al. (2003) and Klein et al. (2010)].
The second method for dealing with computationally fast waves relies on numerical methods that treat these modes in a stable manner. In particular, we focus on methods that integrate the full nonhydrostatic equation set but introduce a splitting strategy to deal with fast wave modes. These approaches are generally referred to as “operator-split methods.” Operator-split methods have been in use for atmospheric models for quite some time, beginning with the first semi-implicit methods of Kwizak and Robert (1971). Since then, semi-implicit methods have been used in atmospheric models at practically all scales (see, e.g., Bonaventura 2000; Giraldo 2005; Restelli and Giraldo 2009). Closely related to semi-implicit methods are split-explicit and fractional step techniques. These methods are similar but instead combine explicit operators, generally splitting on the slow and fast waves. These methods were originally developed by Gadd (1978) for atmospheric models, but they continue to be in use today. The Weather Research and Forecasting model (WRF; Skamarock and Klemp 2008), for example, uses both semi-implicit and explicit splitting, utilizing implicit integration for vertically propagating waves and a split-explicit technique for fast waves in the horizontal.
In this paper, we introduce a new time discretization for models that split the temporal and spatial derivatives using the method of lines (Schiesser 1991; Schiesser and Griffiths 2009). The proposed method offers a simple framework for achieving up to third-order temporal accuracy in a semi-implicit scheme while maintaining computational efficiency. In particular, this approach is designed to outperform a purely explicit formulation for models with a horizontal–vertical aspect ratio greater than 5. Following an operator-split Runge–Kutta–Rosenbrock (RKR) strategy, which combines an explicit Runge–Kutta (RK) method with a linearly implicit Rosenbrock step (Rosenbrock 1963), we obtain a method whose maximum stable time step is constrained only by the horizontal CFL number. To maximize efficiency on parallel systems, the splitting is performed on the horizontal and vertical components of the governing equations so that the implicit solve occurs without requiring off-processor communication (here we assume that vertical columns are not distributed among multiple processors). This strategy differs from the approach of St-Cyr and Neckels (2009), for instance, who apply the implicit solve over the entire domain. In this paper, the RKR time discretization strategy is demonstrated using a high-order finite-volume method in 2D and 3D so as to verify accuracy and stability, but this approach is easily extended to other methods that independently discretize space and time. In particular, discontinuous Galerkin methods, which have been growing in popularity in recent years, would be excellent candidates for use with these time integrators. The approach presented herein is valid for all horizontal scales and hence may be especially well suited for models that utilize adaptively refined meshes with scale differences.
In section 2, we introduce the full nonhydrostatic fluid equations and explain how to incorporate terrain-following coordinates. The RKR discretization is introduced in section 3, wherein we present a first-order, a second-order, and a third-order temporal discretization that is stable for high-order spatial discretizations. We will demonstrate these techniques using high-order finite-volume spatial discretizations, which are explained in section 4, followed by numerical results in section 5. Our conclusions and future work are given in section 6.
2. The nonhydrostatic fluid equations in Cartesian coordinates
List of parameters and physical constants used in this paper.
a. Reference profile splitting
b. Incorporating topography
When topography is present, terrain-following coordinates as introduced by Gal-Chen and Somerville (1975) (GS coordinates) are used to deform the computational domain to match the physical space. However, we do not modify the governing equations from the form (10)–(14) but instead make use of orthonormalization and deorthonormalization operators to accurately compute fluxes in the presence of topography (see section 4d). This approach is analogous to the treatment of edge fluxes that arises on unstructured grids. In addition to simplifying the arithmetic, this approach also avoids problems that may arise from the explicit computation of metric terms (Klemp et al. 2003).
A possible alternative to terrain-following coordinates are so-called shaved-cell methods (see, e.g., Adcroft et al. 1997), which remove the portions of a cell occupied by topography. Unfortunately, if terrain is accurately resolved, this approach will reduce the horizontal extent of an element, and hence the maximum allowable horizontal time step. Modified shaved cells that do not reduce the horizontal extent of cells could also be used, but this technique may significantly degrade the accuracy of the terrain discretization.
As has been shown by Schär et al. (2002), GS coordinates are suboptimal for atmospheric motions, because they tend to introduce spurious grid artifacts above rough terrain. More accurate numerical methods may assist in reducing these errors; in particular, for the tests presented in this paper, these coordinates have been shown to be sufficiently smooth. In the future, this choice of terrain-following coordinate in our model will likely be revisited.
3. RKR schemes
The method of lines approach is one of the most popular methods for constructing high-order finite-volume methods that are applicable to general systems of partial differential equations (PDEs) (Jameson et al. 1981; McCorquodale and Colella 2011). Under this framework, the spatial terms, including the flux and source terms, are discretized first, leading to a system of ordinary differential equations (ODEs) for the state variables within each grid cell. This system is then discretized by means of choosing an appropriate time-stepping scheme. The time-stepping scheme must be chosen so that the eigenvalues of the spatial operator fit within the scheme’s stability region. Explicit schemes are generally computationally inexpensive but possess a restricted stability region, whereas implicit schemes are more costly but possess a large stability region. However, different physical processes can have eigenvalues that have dramatically different structure, and so it may not be appropriate to use a single time-stepping method to integrate all terms of the ODE system.
a. The Runge–Kutta–Rosenbrock approach
Implicit–explicit (IMEX) methods represent a category of general-purpose schemes for ODEs that couple implicit and explicit time integration methods. These methods have been in use as early as the 1970s (e.g., Crouzeix 1980; Varah 1980). More recently, a family of implicit–explicit Runge–Kutta (IMEX-RK) schemes was collected into a general framework by Ascher et al. (1997) in their seminal paper. They showed that it is possible to achieve an essentially arbitrary order of accuracy by correctly interleaving explicit and implicit steps, although with increasing computational expense.
To improve the performance of the IMEX methods, we focus on the family of operator-split RKR methods, which are identical to IMEX schemes, except replacing the computationally expensive implicit step with a so-called Rosenbrock step. Rosenbrock methods were originally developed by Rosenbrock (1963) in the 1960s and later refined by Nørsett and Wolfbrandt (1979). In the atmospheric science community, Rosenbrock-type methods have been used by Lanser et al. (2001) for solving the shallow-water equations on the sphere and adopted by St-Cyr and Neckels (2009) in the development of a fully implicit discontinuous Galerkin mesoscale model. More recently, a framework for high-order RKR time-stepping methods has been presented by Jebens et al. (2011) for use in atmospheric models utilizing cut cells. These methods are also popular in atmospheric chemistry modeling (Sandu et al. 1997; Verwer et al. 1999), because reaction equations tend to operate on very fast time scales.
When applied to systems of ODEs obtained from time-split PDE systems, the Newton–Krylov method is usually initialized by taking x(0) to be the value of x obtained at the previous time step. Although this method converges quadratically, the Jacobian 
b. Computation of the Jacobian
The main cost of the Rosenbrock method is in the implicit vertical step, which consists first of the construction of the Jacobian matrix and second of the matrix solve. Under our current implementation, the matrix solve is handled by the Linear Algebra Package (LAPACK) banded matrix solver. These routines are roughly twice as fast as the general matrix solver routines but nonetheless contribute significantly to the total wall-clock time of the implicit solve. No additional effort has been made to optimize these routines for our problem. The bulk of optimization efforts have instead focused on the construction of the Jacobian matrix.
If the function g is simple enough, it may be possible to instead formulate the Jacobian analytically. This strategy significantly reduces the computation time required for constructing the Jacobian and so will be our method of choice in this paper. Under our finite-volume formulation, a quasi-linear flux function in the vertical (see section 4f) is chosen in order to facilitate an easy formulation of the Jacobian matrix. It will be shown (see section 5a) that the effort in formulating an analytic Jacobian significantly reduces computation time for the implicit solve. The analytic expressions for each term of the Jacobian have been omitted here because they are quite lengthy.
c. A crude splitting scheme
d. The Strang carryover scheme
e. The Ascher–Ruuth–Spiteri (2, 3, 3) scheme
This scheme is linearly third-order accurate in both f and g and any cross-terms that arise from the integration procedure, but it is only nonlinearly third-order accurate in f. In fact, when g = 0 the stability region for this scheme is exactly the stability region of the usual three-stage third-order-accurate Runge–Kutta operator. This scheme requires three explicit steps per time step and two Rosenbrock steps, with each Rosenbrock step consisting of a single evaluation of the Jacobian and a single linear solve. As a consequence, the overall computational cost of this method is approximately twice that of the Strang carryover scheme.
4. Spatial discretization
In this section, we turn our attention to the spatial discretization of the 3D nonhydrostatic governing equations (10)–(14) using a high-order finite-volume scheme. Here, we describe the key components of our algorithm for discretely evolving the state vector over time.
a. Finite-volume approach
For each explicit substage, the 3D finite-volume algorithm proceeds as follows:
Construct the left and right edge-averaged state vector from the subgrid-scale reconstruction.
Deconvolve the edge averages to give a fourth-order-accurate edge-centered approximation of the state vector.
Transform vector quantities into the orthogonal frame at the edge center point.
Compute the pointwise flux across the edge using a Riemann solver.
Transform vector fluxes into the Cartesian frame using the deorthonormalization matrix.
Convolve the pointwise fluxes to obtain a fourth-order-accurate edge-averaged approximation of the flux.
Compute element-averaged source terms.
Update the element-averaged state vector within each element.
b. The subgrid-scale reconstruction
It should be emphasized the aforementioned reconstructions produce either fifth-order-accurate (using the five-point stencil) or third-order-accurate (using the three-point stencil) approximations to the edge-averaged state vector. If we were to directly evaluate the flux using these reconstructed edge averages, the resulting edge-averaged flux would only be second-order accurate. This result arises because the flux function is not a linear operator, and so the average of the flux is not the flux of the average. Hence, without further application of a convolution/deconvolution operator, both of these reconstructed edge averages will lead to a scheme that is formally second-order accurate. Nonetheless, the stencil (51) and (52) generally produces less diffusive and more accurate results, whereas using the three-point stencil (53) and (54) reduces the bandwidth of the Jacobian matrix and so generally results in a faster vertical solve.
c. Fourth-order convolution and deconvolution operators in 3D
In our treatment, the edge averages in the vertical direction are used as direct input to the Riemann solver. This implies that this method is only second-order accurate in the vertical. However, these errors are typically much smaller than the associated truncation errors in the horizontal for two reasons: First, in operational global atmospheric models the grid spacing in the vertical is typically much smaller than the grid spacing in the horizontal. Second, vertical wind speeds are typically much less than the corresponding horizontal wind speeds.
d. Orthonormalization
e. The AUSM+-up Riemann solver
The Advection Upstream Splitting Method (AUSM+-up) approximate Riemann solver of Liou (2006) was recently developed with the goal of enhancing the accuracy of the Riemann solution in the low Mach number regime. Many other commonly used Riemann solvers, including the popular solver of Rusanov (1961) and the solver of Roe (1981), do a poor job in the very low Mach number regime because they introduce a significant amount of numerical diffusion that can smear out the solution (see Ullrich et al. 2010). Here, we give a short overview of the algorithmic implementation of this solver without delving into the mathematical details.
In general, standalone Riemann solvers require that the velocity components of the input state vector be written in an orthogonal frame. Orthonormalization is performed by multiplying the velocity vector by the orthonormalization matrix, which is described in section 4d and yields velocity components (υ⊥, υ1, υ2) in the orthogonal frame. The momentum flux, which is computed in the orthogonal frame, must similarly be transformed back into the Cartesian frame, which can be obtained by multiplying the Riemann flux by the inverse of the orthonormalization matrix.
f. The modified AUSM+-up Riemann solver
g. Nonreflecting boundary conditions
h. Explicit diffusion
Although the Riemann solver provides a mechanism for maintaining stability via the addition of implicit diffusion, in some circumstances it may be desirable to specify an explicit viscous forcing. Traditionally, a parameterization of the viscosity appears in the form of a Laplacian-type viscous term on the right-hand side of the nonhydrostatic equations of motion (10)–(14). However, as argued by Giraldo and Restelli (2008), more physical parameterizations of dynamic viscosity exist. Nonetheless, for simplicity we will use this form of diffusion when needed for selected test cases in section 5.
5. Numerical results
In this section, we present a selection of 2D (x–z) and 3D numerical results in order to verify the convergence and accuracy properties of the schemes discussed in this paper. In section 5a, we look at a rising thermal bubble in order to verify that our scheme is consistent with other models and to show the effect of the first-, second-, and third-order-accurate time-stepping schemes. In this section, we also compare the three- and five-point vertical stencils and show the timing results for each of the available model configurations. In section 5b, we study the spatial and temporal convergence properties of these schemes using a density current test case with explicit viscous forcing. The performance of our model for flow over topography is studied in sections 5c and 5d. The latter case is used to verify stability of the time integrators, even for a large horizontal–vertical aspect ratio. The problem of a 3D geostrophically balanced flow in a channel is studied in section 5e, again using a large horizontal–vertical aspect ratio. This test case is used to verify fourth-order horizontal convergence of our numerical method. Finally, in section 5f, we look at the evolution of a baroclinic wave in the channel on both a constant f plane and a constant β plane. These tests further evaluate our scheme on a wide range of possible scales, ranging from the microscale with the rising thermal bubble test to the global scale with the baroclinic instability.
a. Rising thermal bubble
The 2D (x, z) rising thermal bubble test case is essentially ubiquitous in the study of nonhydrostatic mesoscale models. This test follows the evolution of a warm bubble in a constant potential temperature environment. The warm bubble leads to a positive perturbation in the vertical velocity field, which acts to carry the bubble upward. As the bubble moves upward, shearing quickly deforms the circular bubble into a mushroom cloud. Here, we follow the initialization procedure described by Giraldo and Restelli (2008), which is a variation of the bubble experiments of Robert (1993). Because no explicit diffusion is added to this test, we do not anticipate that the solution will converge as spatial resolution is refined. However, at finer resolutions we do observe more finescale features of the thermal bubble, including tighter winding of the trailing edges at later times and sharper spatial gradients. Nonetheless, our comparisons for this test case are purely qualitative.
The potential temperature perturbation at t = 700 s with crude splitting is plotted in Fig. 1. Four different resolutions ΔX = ΔZ are shown that range between 20 and 2.5 m. At even the finest resolution, this scheme performs exceptionally poorly, unable to even resolve the correct convection velocity at low resolutions or the anticipated winding of the bubble’s leading edges at higher resolutions. This approach is equivalent to low-order time splitting of the horizontal and vertical motions; its poor behavior may be indicative of problems with other low-order time-split approaches.
Plots of the potential temperature perturbation for the rising thermal bubble test case with crude splitting and three-point vertical stencil at time t = 700 s and four choices of resolution. The time step is chosen to be 0.05 s at the coarsest resolution and is otherwise proportional to the grid spacing. Contour lines are from 300 to 300.5 K with a contour interval of 0.05 K. The 300-K contour line is shown in light gray to emphasize numerical oscillations due to undershoots and overshoots.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
The Strang carryover and ARS(2, 3, 3) schemes perform significantly better. The results with these two methods are plotted in Fig. 2 for the three-point vertical stencil (53) and (54) and in Fig. 3 for the five-point vertical stencil (51) and (52). These plots show both time-stepping schemes on the same axes, because the symmetry of the bubble allows us to only plot half the domain. As a consequence, we can clearly see differences between these two time-stepping schemes in the final results. The plots are again made at four resolutions after 700 s. In both sets of plots, the ARS(2, 3, 3) scheme appears to perform better than the Strang carryover scheme, exhibiting slightly more winding and enhanced gradients, especially at lower resolutions. The results from using the five-point stencil are observably better than those with the three-point stencil, and so this wider stencil may be desirable in improving accuracy in real applications. Both of these schemes match the results reported in Giraldo and Restelli (2008) very closely.
As in Fig. 1, but showing (left) the Strang carryover scheme and (right) the ARS(2, 3, 3) scheme on the same axes with three-point vertical stencil given by (53) and (54). We have exploited the symmetry of the rising bubble to plot these schemes side by side for comparison.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
As in Fig. 2, but using the five-point vertical stencil given by (51) and (52), showing (left) the Strang carryover scheme and (right) the ARS(2, 3, 3) scheme on the same axes.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
Timing results for the rising thermal bubble test on a grid with 20-m horizontal and vertical resolution with Δt = 0.05 s are given in Table 2 for a variety of model configurations. We have tested out the Strang carryover and ARS(2, 3, 3) schemes with either a numerical or an analytic Jacobian, either a three-point or a five-point vertical stencil, and either a Rosenbrock solve or a fully implicit solve in the vertical. The timing results from the crude splitting scheme were not reported here because they very closely match the timing results from the Strang carryover scheme. As a baseline, we have run a fully explicit scheme that uses the RK3 method of Gottlieb et al. (2001) for the full set of nonhydrostatic equations. We observe that the cheapest RKR method is the Strang carryover scheme with analytic Jacobian, three-point vertical stencil, and Rosenbrock solver. This scheme is just over twice as expensive as the corresponding explicit method. The more accurate five-point stencil adds an overhead of approximately 25%. The fully implicit method approximately doubles the computation time, implying that roughly twice as many implicit solves are computed at each time step, but does not significantly increase the accuracy of the results. The ARS(2, 3, 3) scheme leads to a similar overhead, because it explicitly requires two implicit solves per time step.
Timing results from the rising thermal bubble test on a 50 × 50 grid (20-m resolution) with Δt = 0.05 s until t = 700 s using a variety of model configurations. Timing is normalized to the fastest configuration, which uses an explicit RK3 method for time stepping in both the horizontal and vertical. On a recent MacBook Pro with 2.2-GHz Intel Core i7 chip and two processors, this parallel configuration required 32 s to run.
b. Straka density current
The results of our model with the Strang carryover and ARS(2, 3, 3) schemes are plotted at t = 900 s in Fig. 4 for the three-point vertical stencil and in Fig. 5 for the five-point vertical stencil. Four resolutions ΔX = ΔZ are shown. As with the rising thermal bubble, we observe a very slight improvement from using the ARS(2, 3, 3) scheme, which is more prominent at lower resolutions. Also, we observe a significant improvement at lower resolutions when using the five-point vertical stencil with both time-stepping schemes. The appearance of three well-defined Kelvin–Helmholtz rotors is not obvious until 50-m resolution with the three-point vertical stencil but is clear at 100-m resolution with the five-point vertical stencil. Both methods otherwise perform well with grid refinement.
Plots of the potential temperature perturbation for the Straka density current test case with three-point vertical stencil at time t = 900 s and four choices of resolution. The time step is 0.5 s for a grid spacing of 200 m and scales with the spatial resolution. Contour lines are from 291 to 300 K with a contour interval of 1 K. The 300-K contour line is shown in light gray to emphasize numerical oscillations. At each resolution, we show (top) the Strang carryover scheme and (bottom) the ARS(2, 3, 3) scheme for comparison.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
As in Fig. 4, but with a five-point vertical stencil. Again, we plot (top) the Strang carryover scheme and (bottom) the ARS(2, 3, 3) scheme for comparison.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
Spatial convergence plots showing the L2 difference in the density perturbation, horizontal velocity, vertical velocity, and potential temperature perturbation as a function of resolution and three-point vertical stencil and using the three-point vertical stencil. The solid line denotes the fully explicit scheme with RK3 time step, whereas the dashed line and dotted–dashed line correspond to the Strang carryover and ARS(2, 3, 3) schemes, respectively. The time step at 200 m is 0.5 s and is otherwise proportional to the grid spacing so as to maintain a constant CFL number. The gray lines denote perfect (top) first- and (bottom) second-order scaling rates as a function of grid spacing.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
As in Fig. 6, but for the five-point vertical stencil.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
We observe significantly different convergence behavior results for each of the four choices q. In fact, the fully explicit method, Strang carryover method, and ARS(2, 3, 3) method all seem to behave similarly when purely considering the perturbation in the potential temperature. Because the potential temperature perturbation is not a prognostic variable, its L2 errors are a function of errors in both the density perturbation and the potential temperature density perturbation. In computing (104), these errors approximately cancel and so lead to very similar behavior among all time-stepping schemes. However, for each of the variables ρ′, u, and w and for (ρθ)′ (not shown) the observed errors are significantly different between the Strang carryover and ARS(2, 3, 3) schemes. Although the Strang carryover scheme generally performs more poorly with the three-point vertical stencil, its spatial convergence rate roughly matches that of the other schemes. However, with the five-point vertical stencil (Fig. 5), we observe that the Strang carryover scheme seems to converge at least one order of accuracy more slowly than the other two schemes. The ARS(2, 3, 3) scheme closely mirrors the fully explicit scheme, suggesting better overall performance of this method.
The results from performing a temporal convergence analysis on this problem are given in Fig. 8. In this case, we set ΔX = ΔZ = 100 m = constant and adjust the time step size Δt to observe the rate of temporal convergence for each of the methods. The error norms are obtained by computing the L2 norm (104) with a reference solution obtained by running the test case with ΔX = ΔZ = 100 m and Δt = 1.5625 × 10−2 s. For the density perturbation, horizontal velocity, and vertical velocity, we observe second-order convergence in Δt for the Strang carryover scheme and near-third-order convergence in Δt for the ARS(2, 3, 3) scheme. The potential temperature perturbation again shows different behavior because of cancelation in the computation of the L2 errors but nonetheless yields improved errors under the ARS(2, 3, 3) scheme.
Temporal convergence plots showing the L2 error in the density perturbation, horizontal velocity, vertical velocity, and potential temperature perturbation as a function of time step Δt for a fixed grid spacing of ΔX = 100 m. The solid line and dashed line correspond to the Strang carryover and ARS(2, 3, 3) schemes, respectively. The gray lines denote perfect (top) first- and (bottom) second-order scaling rates as a function of time step.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
c. Schär mountain
The Schär mountain profile with ΔX = ΔZ = 500 m and magnified such that (x, z) ∈ [−10 000, 10 000] × [0, 4000] m.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
This test case is run at ΔX = 250 m and ΔZ = 210 m using the Strang carryover scheme and five-point vertical stencil. The observed horizontal and vertical wind speeds after 10 h are plotted in Fig. 10. The fields are smooth and agree well with published results in Klemp et al. (2003) and Giraldo and Restelli (2008). In particular, we do not observe any artifacts from the inhomogeneous terms described by Klemp et al. (2003), which would introduce spurious noise at coarse grid resolutions. As expected for steady-state solutions, the ARS(2, 3, 3) scheme produces qualitatively similar results.
Steady-state flow over the Schär mountain after 10 h with ΔX = 250-m- and ΔZ = 210-m resolution, 5-point vertical stencil, and Strang carryover scheme. The simulation is run to t = 10 h with a time step of Δt = 0.6 s. (a) The horizontal velocity has contour values between 8 m s−1 and 12 m s−1 with a contour interval of 0.2 m s−1 with emphasis on the 10 m s−1 contour. (b) The vertical velocity has contour values between −2 and 2 m s−1 with a contour interval of 0.05 m s−1 with emphasis on the 0 m s−1 contour. In (a), values less than 10 m s−1 are shaded. In (b), negative values are shaded.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
d. Flow over an isolated mountain
The simulation domain is taken to be (x, Z) ∈ [−14ac, 14ac] × [0, 21 000] m, and the time step is chosen to be 
The results of these simulations using the Strang carryover time-stepping scheme are plotted in Fig. 11 for four choices of ac: ac = 1 km (Fig. 11a), ac = 10 km (Fig. 11b), ac = 100 km (Fig. 11c), and ac = 1000 km (Fig. 11d). We observe agreement with the results of Dudhia (1993) in all four flow regimes, suggesting our model is correctly capturing the dynamics of these regimes. Further, our model appears stable even with a vertical CFL number of nearly 500 as in Fig. 11d. The total computation time is also observed to be roughly equivalent in all cases.
Plots of vertical velocity for the Dudhia (1993) mountain test case with background flow speed u = 10 m s−1 and mountain half-width ac given by (a) 1, (b) 10, (c) 100, and (d) 1000 km. Grid spacing is taken to be ΔX = ac/5 and ΔZ = 420 m, leading to a maximum aspect ratio in (d) of ΔX/ΔZ = 476. The simulation is run up to t = 21.6ac/u with a fixed time step of 
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
e. Steady-state geostrophically balanced flow in a channel
Error norms are given in Table 3 for an f-plane approximation and Table 4 for the β-plane approximation. The simulations are run with a variable horizontal resolution of 400, 200, 100, and 50 km and a uniform vertical resolution of 1 km (30 equally spaced vertical levels) for one day. The time step at 400-km resolution is 960 s and otherwise scales with resolution. We observe convergence that is slightly less than fourth order for the vertical momentum field and slightly better than fourth order for the potential temperature field. Because hydrostatic balance is guaranteed by the background splitting technique described in section 2, errors are only accumulated because of an imbalance in the geostrophically balanced components, and hence increasing the number of vertical levels does not have a significant impact on the error norms. Discrepancies in these errors at each vertical level trigger the slight imbalances in the vertical velocity. Because this is a steady-state test case, the errors in this analysis are dominated by errors in the spatial reconstruction, and so similar error norms are observed with both the crude and ARS(2, 3, 3) time-stepping schemes.
Relative errors in the vertical momentum field ρω and potential temperature density field ρθ for the geostrophically balanced flow in a channel test with an f-plane approximation and Strang carryover time-stepping scheme. Errors are represented in scientific notation using the form a (b) for mantissa a and exponent b, under base 10. A convergence study is performed by varying the horizontal resolution. The computed order of accuracy is obtained from a least squares fit through the data.
f. Baroclinic instability in a channel
The setup resembles the baroclinic wave experiments on the sphere suggested by Jablonowski and Williamson (2006). The unbalanced perturbation acts as a trigger for baroclinic waves that grow explosively over a 10–12-day simulation period. Such a flow is characteristic for the midlatitudes. The channel test thereby assesses how well the finite-volume scheme simulates large-scale flow fields with large aspect ratios. All simulations are run with the ARS(2, 3, 3) scheme and utilize a 100-km horizontal grid spacing with 30 equally spaced vertical levels and a model top at 30 km. The Strang carryover scheme yields results that are qualitatively similar at this resolution. Again, no explicit viscosity or sponge layer is used. The time step is Δt = 240 s. Timing results for this test case are given in Table 5 and demonstrate the benefits of using the RKR methods over a purely explicit approach for this problem.
Timing results from the baroclinic instability in a channel on an 80 × 12 × 30 grid (ΔX = 500 km, ΔZ = 1 km) and three-point vertical stencil. Timing is normalized to the fastest configuration, which uses the Strang carryover scheme and analytic Jacobian. On a recent MacBook Pro with 2.2-GHz Intel Core i7 chip and two processors, this parallel configuration required 6.8 s day−1.
Snapshots of the simulation for the f-plane approximation at day 12 are plotted in Fig. 12. The figure depicts the horizontal cross sections of the pressure, temperature, and relative vorticity at 500 m. This vertical position corresponds to the height of the lowermost model level. We observe that the baroclinic wave has almost broken, which takes place around day 13.5. The flow has formed distinct low and high pressure systems that are associated with sharp temperature fronts and sharp gradients in the relative vorticity field.
Simulation results from the baroclinic instability in a channel computed at day 12 using the ARS(2, 3, 3) scheme with the f-plane approximation. The simulation is run at a horizontal resolution of 100 km and a vertical resolution of 1 km with a time step of 240 s. Contour lines are as indicated on each plot. The 942-hPa line is enhanced in the pressure plot. The zero line in the relative vorticity plot is enhanced, and negative values are plotted using dashed lines.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
The corresponding simulation results on the β plane are plotted in Fig. 13. Here, we show the identical fields, but now at day 10 before wave breaking events set in (around day 11). The presence of the planetary vorticity gradient has sped up the evolution of the baroclinic wave. Again, the low and high pressure systems are connected to sharp frontal zones in the temperature and vorticity fields that resemble realistic flow conditions. It is interesting to note that the β-plane simulation leads to a more confined flow field that has not spread to the northern and southern edges of the domain by day 10. These differences between the f-plane and β-plane simulations will be discussed in greater detail in C. Jablonowski et al. (2011, unpublished manuscript) alongside a comparison to other nonhydrostatic channel models. The main focus of this test is to demonstrate that the RKR schemes reliably simulate the evolution of atmospheric flow fields that are relevant for the large (midlatitudinal) portion of global atmospheric general circulation models. To verify correctness of our results, this test has also been run using the model of Norman et al. (2011) and qualitatively similar results were observed.
As in Fig. 12, but for the β-plane approximation at day 10. The 943-hPa line is enhanced in the pressure plot. The zero line in the relative vorticity plot is enhanced, and negative values are plotted using dashed lines.
Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-10-05073.1
6. Conclusions
In this paper, we have presented a new approach for discretizing the nonhydrostatic Euler equations in Cartesian geometry using high-order finite-volume methods and a horizontal–vertical splitting strategy based on Runge–Kutta–Rosenbrock (RKR) time integration schemes. For atmospheric problems where the vertical grid spacing is usually much smaller than the horizontal, this strategy allows us to simulate the full Euler equations while only constraining the time step by the horizontal grid spacing. We have presented time-stepping schemes based on a crude strategy, a scheme that uses a Strang splitting and carryover strategy, and a higher-order scheme based on an approach attributed to Ascher et al. (1997). These time-stepping schemes have been implemented in a mesoscale atmospheric model that utilizes a high-order finite-volume-based approach. The crude time-stepping scheme is shown to be highly diffusive for thermal bubble experiments and shows no benefit over the Strang carryover scheme, which requires the same number of explicit and implicit steps per time step. The ARS(2, 3, 3) scheme shows a mild improvement over the Strang carryover approach but requires two implicit steps per time step. However, the higher-order accuracy in time this scheme affords may be desirable. Further, we compared a three-point and a five-point vertical reconstruction stencil within the model and observed significantly better results with the five-point stencil, but with a slight computational overhead.
Numerical results have shown our approach to be accurate, stable, and applicable to a range of atmospheric flows and horizontal–vertical aspect ratios. By using a fourth-order reconstruction strategy in the horizontal, we observe clear fourth-order convergence for flows with a small vertical velocity. Horizontal–vertical aspect ratios up to 500:1 have been tested under our scheme and verified to be stable up to a horizontal CFL number of 1.0.
The results in this paper suggest that our horizontal–vertical dimension splitting strategy is a promising option for any high-order finite-volume or discontinuous Galerkin-based method. This model has already been extended to a full global nonhydrostatic dynamical core on a cubed-sphere grid utilizing high-order finite-volume methods in Ullrich and Jablonowski (2012, manuscript submitted to J. Comput. Phys.).
Acknowledgments
The authors would like to acknowledge the contributions of Steve Ruuth, whose fruitful comments helped in developing the Strang carryover approach for this paper. We would also like to thank Matthew Norman for his work implementing the baroclinic instability test and providing a comparison for our results. Finally, we thank the two anonymous reviewers for their very helpful commentary in improving this work. This work was supported by the Office of Science, U.S. Department of Energy, Award DE-SC0003990 and a University of Michigan Rackham Predoctoral fellowship.
APPENDIX
Converting between η and z Coordinates
Here, Φ and T are given by (109) and (112), respectively. The starting value of η0 = 10−7 is used for all Newton iterations, corresponding to a model top of about 100 km. If a higher model top is required, the value of η0 needs to be decreased. Convergence is deemed to have occurred if |ηn+1 − ηn| ≤ 10−14 and usually takes about 10 iterations in most cases.
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