## 1. Introduction

A main effect of acoustic waves in the atmosphere is a rapid adjustment to a balanced state with respect to pressure–density perturbations. As long as one is not interested in this adjustment process itself but rather in atmospheric dynamics on comparatively longer time scales, it is often attractive to filter acoustic waves from the dynamic equations. Thus, simplified (i.e., soundproof) equations enable a focus on the dynamics of more significant mesoscale processes, while the absence of fast waves also allows for larger time steps in numerical integrations. A classic example is the Boussinesq system, where the prognostic continuity equation is replaced by a diagnostic divergence constraint on the wind field. This yields a good approximation of atmospheric dynamics on spatial scales smaller than the atmospheric-scale height. Deep convection and wave propagation, however, experience effects of the vertical decrease of ambient density that cannot be reproduced in the Boussinesq approximation. This density effect causes, for example, gravity waves—typically generated in the troposphere—to grow in amplitude until they get unstable, break, and finally dissipate and interact with the large-scale flow in the middle atmosphere (Lindzen 1981; Fritts and Alexander 2003). To include such effects, two more general approaches have been proposed. One is the well-known anelastic equations first introduced by Batchelor (1953) and Ogura and Phillips (1962) and further developed by Lipps and Hemler (1982) and Lipps (1990). They use a more general divergence constraint, which takes the decreasing background density into account as a weighting factor for the wind. A potential drawback of the anelastic system is that it requires the leading-order background atmosphere to be close to isentropic. Some indications exist that corresponding errors might often be negligible even for realistic tropospheric stratifications (Smolarkiewicz and Dörnbrack 2008; Klein et al. 2010; Smolarkiewicz and Szmelter 2011). However, an approach that does not constrain the background potential temperature as much as in the anelastic equations is offered by the pseudo-incompressible system (Durran 1989; Durran and Arakawa 2007; Durran 2008). By introducing the so-called pseudo-incompressible density, the compressible coupling between pressure and density fluctuations and thus the transfer from elastic potential to kinetic energy is cut off. Smolarkiewicz and Dörnbrack (2008) underlined that the Durran system retains the full momentum equation and, consequently, admits unabbreviated baroclinic production of vorticity, as also observed by Klein (2009). Achatz et al. (2010) demonstrated a multiscale asymptotic consistency between the pseudo-incompressible system and the compressible Euler equations to leading order, which cannot be observed for the anelastic system.

The pseudo-incompressible equations are thus suitable for studying full-scale gravity wave dynamics from their generation until their breaking. In this problem the distances covered by waves of interest, both in the horizontal and in the vertical, are typically far too large to allow an explicit direct simulation of the generated turbulence. For its parameterization an implicit large-eddy-simulation (ILES) method as developed by Adams et al. (2004) and Hickel et al. (2006, 2007) seems attractive. This so-called adaptive local deconvolution method (ALDM) represents a merger of numerical method and subgrid-scale (SGS) turbulence parameterization. Instead of keeping numerical truncation errors small, which can be quite costly or unfeasible, and later dissipate the solution with an explicit turbulence model, numerical truncation is deliberately exploited and tuned to act as turbulence model. This holistic approach results in a particularly reliable and efficient method, while turbulence spectra are matched well. ALDM has been calibrated to be consistent with eddy damped quasi-normal Markovian (EDQNM) spectral turbulence theory (Hickel et al. 2006). Indeed, comparisons with direct numerical simulation (DNS) data of turbulent flows showed better spectral behavior than standard turbulence models. ALDM was originally developed for incompressible flows (Hickel et al. 2006, 2007) and subsequently extended to fully compressible turbulence (S. Hickel 2011, unpublished manuscript; Hickel and Larsson 2008). Although its already widespread application in engineering applications (Klar et al. 2011; Hickel et al. 2011) had been limited to the description of the effects of unstratified turbulence, recent results by Remmler and Hickel (2012a,b) demonstrate that the method can also be applied to stratified turbulence governed by the Boussinesq equations. Here we report the first application of ALDM to full-scale atmospheric flows beyond the Boussinesq approximation.

For this purpose a novel fully conservative discretization method for Durran’s pseudoincompressible system has been developed, as previously achieved, to the best of our knowledge, only by Smolarkiewicz and Dörnbrack (2008) and Smolarkiewicz and Szmelter (2011). As the implicit turbulence model of ALDM is fully merged with the spatial discretization, an extension to another system of partial differential equations can be challenging. This paper shows how these challenges have been resolved within the framework of our newly developed atmospheric flow solver Pseudo Incompressible Flow Solver with Implicit Turbulence (pincFloit) model. We emphasize that many numerical issues discussed here, such as the convergence criterion for the pseudo-incompressible equations and a divergence correction for reducing the number of Poisson-solver iterations, are relevant to any kind of finite-volume method.

The paper is organized as follows. In section 1 we recall the pseudo-incompressible equations and present them in a conservative and scaled form suitable for the discretization by a conservative finite-volume model. The general approach to the discretization is given in section 2. Some special issues concerning pitfalls in the application of conservative finite-volume methods to stratified fluids with the pseudo-incompressible divergence constraint are discussed in section 3. Numerical test cases in section 4 close the text, while section 5 gives a summary and conclusions.

### a. Governing equations

^{1}Fr, and the Reynolds number Re; the velocity

**u**= (

*u*,

*υ*,

*w*) with its zonal, meridional, and vertical components, respectively; and the potential temperature

*θ*and the pseudo-incompressible (effective) density

*ρ*. Further quantities are the Exner pressure

*κ*=

*R*/

*c*, where

_{p}*c*is the specific heat at constant pressure and

_{p}*R*is the specific gas constant of air; a constant reference pressure

*p*

_{0}; and the pressure

*p*. Following the notation of Klein (2009), we introduce

*γ*=

*c*/

_{p}*c*set to 1.4 in all our calculations. The pseudo-incompressible (effective) density is defined through the equation of state

_{V}^{2}satisfies

**Π**is given by

*δ*and the dynamic shear viscosity coefficient

_{ij}*η*. The effect of volume viscosity is not considered. Note that the viscosity term is responsible for the resolved grid-scale dissipation and is part of the model even if an implicit LES is used since the latter treats the subgrid-scale dissipation.

### b. Transport of potential temperature

*ρθ*is constant in time. Note that for diabatic flows a heat source enters the RHS of the divergence constraint and the transport equation of the potential temperature.

## 2. Discretization with ILES

We present the implicit LES proposed by S. Hickel (2011, unpublished manuscript), Hickel et al. (2006, 2007), and Hickel and Larsson (2008) named ALDM. Originally introduced for incompressible and fully compressible flows, we adopt the approach to the pseudo-incompressible equations. We present general ideas in the main part and put some of the more technical aspects into the appendices.

### a. General setup

All variables are stored in a C-grid fashion. Figure 1 shows a finite-volume cell of the density with the velocities defined at the cell interfaces. Each momentum component has its own cell shifted with respect to the mass cell by half a cell in the corresponding direction. To calculate the fluxes across the cell surfaces, point values of velocity and scalars are reconstructed there. To denote the reconstructed value at the right wall of cell *ijk* we use the symbol *L*, *F*, *B*, *U*, and *D* for reconstruction to the left, forward, backward, upward, and downward cell interface, respectively; please see also Fig. 2. The equations for these reconstructions terms are given in the appendices.

Position for values from reconstruction in zonal, meridional, and vertical direction.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Position for values from reconstruction in zonal, meridional, and vertical direction.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Position for values from reconstruction in zonal, meridional, and vertical direction.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

### b. Adaptive deconvolution (reconstruction)

To obtain the values at the cell interface, ideas from numerical gas dynamics are borrowed and extended. In the weighted essentially nonoscillatory (WENO) method (Shu 1997) data within a cell is reconstructed using the cell averages from neighboring cells as depicted in Fig. 3 on the example of a fifth-order WENO scheme with third-order reconstruction polynomials. Note that the parabolas do not interpolate the data points since they are finite-volume cell averages and not finite-difference point values. At the cell interface *x _{i}*

_{+1/2}three reconstructed values exist from the three stencils. In WENO the reconstructed values are averaged using solution adaptive weights. These weights are chosen in such a way that the order of accuracy in smooth flows approaches the maximum for the given stencil width while unphysical oscillations are avoided by keeping the total variation in nonsmooth regions bounded.

Three possible reconstructions of *φ* in cell *i* with quadratic functions from the cell-centered values *φ*.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Three possible reconstructions of *φ* in cell *i* with quadratic functions from the cell-centered values *φ*.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Three possible reconstructions of *φ* in cell *i* with quadratic functions from the cell-centered values *φ*.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

ALDM differs from WENO in two ways. First, with ALDM the weights in the convex combination of the reconstruction polynomials are used to optimize the nonlinear spectral numerical dissipation that acts as an implicit SGS model rather than maximizing the formal order of accuracy. Second, in contrast to WENO, where polynomials of a certain degree *n* are used to obtain a convergence order 2*n* − 1, in ALDM first-, second-, and third-order reconstructions are blended. The blending weights, again, are degrees of freedom that are used for SGS model tuning. For more details please see the appendices or Hickel et al. (2006, 2007).

### c. Numerical flux function

*i*,

*j*,

*k*) we need to discretize the divergence operator, which is done in a conservative finite-volume way:

*f*,

*g*, and

*h*indicate the fluxes in the three spatial directions as depicted in Fig. 4 for the flux of zonal momentum. The numerical flux function

*f*

_{i}_{+1/2,j,k}is the second ingredient of ALDM for ILES. It consists of a central term of high order and an artificial viscosity term. Mass and momentum have slightly different flux functions.

Location of the *ρu*-momentum fluxes with respect to the staggered *ρu*-momentum finite-volume grid (gray) and the mass grid (black).

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Location of the *ρu*-momentum fluxes with respect to the staggered *ρu*-momentum finite-volume grid (gray) and the mass grid (black).

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Location of the *ρu*-momentum fluxes with respect to the staggered *ρu*-momentum finite-volume grid (gray) and the mass grid (black).

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

#### 1) Mass flux function

*u*, which satisfies the pseudo-incompressible divergence constraint. The second term is a dissipative term, with a diffusion coefficient given in (13), which itself depends on the roughness of the velocity field and a tuning parameter

*σ*. In a smooth velocity field a density distribution is (almost) not diffused numerically with this SGS ansatz. Throughout our calculations we use for the tuning parameter

_{C}#### 2) Momentum flux function

*ρu*

_{i}_{,j,k}. The interpolation of the density

*u*. A list of all flux components can be found in appendix A. Also note that

*σ*

^{u}depends on the location if the grid is not uniform—that is, with varying mesh size (Hickel et al. 2006). In pincFloit a uniform Cartesian mesh is used so that this parameter is a constant

## 3. Numerical issues

The implicit LES was devised for incompressible flows by Hickel et al. (2006) and for fully compressible flows by Hickel and Larsson (2008) and S. Hickel (2011, unpublished manuscript). The application of ALDM to a model with atmospheric background stratification and a background density rapidly decaying in the vertical lead to a number of problems—the solution of which we discuss in this section.

### a. Flux function: Consistency between continuity and momentum equation

So far we have not devised a rule how to interpolate the densities

*t*=

*t*:

_{n}*j*and

*k*. The density at the zonal-velocity position is taken to be

*ρ*

_{i}_{+1/2}= (

*ρ*+

_{i}*ρ*

_{i}_{+1})/2. Note that for constant

*u*the correction term containing

*σ*is absent. The position of density, velocity, and fluxes can be seen in Fig. 5.

Position and indices for density, zonal velocity, and momentum flux.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Position and indices for density, zonal velocity, and momentum flux.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Position and indices for density, zonal velocity, and momentum flux.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

*u*=

_{i}*u*= const, and obtain

*i*reads

*i*+ 1

*ρ*

_{i}_{+1/2,j,k}

*u*

_{i}_{,j,k}. We now require that momentum transport [(18)] and mass transport [(9)] are equal:

*j*and

*k*for completeness. This interpolation rule, which is the only possible solution of the form

*a*

_{−∞}=

*b*

_{−∞}=

*a*

_{∞}=

*b*

_{∞}= 0, is illustrated in Fig. 7. This reasoning can be generalized to 3D. A list of all interpolation rules can be found in appendix B.

The two density differences symbolized with arrows have to be equal to achieve consistency between momentum and effective continuity equation. The reconstructed values of the ALDM densities

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

The two density differences symbolized with arrows have to be equal to achieve consistency between momentum and effective continuity equation. The reconstructed values of the ALDM densities

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

The two density differences symbolized with arrows have to be equal to achieve consistency between momentum and effective continuity equation. The reconstructed values of the ALDM densities

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Illustration of density interpolation rule (21). Position of density *f ^{ρu}* (filled circle) and reconstructed densities for mass flux

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Illustration of density interpolation rule (21). Position of density *f ^{ρu}* (filled circle) and reconstructed densities for mass flux

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Illustration of density interpolation rule (21). Position of density *f ^{ρu}* (filled circle) and reconstructed densities for mass flux

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

### b. Temporal discretization

#### 1) Choosing the time step

*ν*is set to 0.9 in most of the calculations in conjunction with third-order Runge–Kutta schemes.

*a*must be considered as well (S. Remmler, TU Munich, 2010, personal communication):

*t*leads to

*u*

_{max}= max|

*u*| and

*a*

_{max}= max|

*a*|, then calculate the time step according to the standard CFL condition Δ

*t*

_{CFL}= min(

*ν*Δ

*x*/|

*u*|, Δ

*t*

_{max}) and if the speedup Δ

*u*=

*a*

_{max}Δ

*t*

_{CFL}is essential (i.e., if Δ

*u*>

*ɛu*

_{max}with e.g.,

*ɛ*= 10

^{−2}), then compute

*N*. A sensible time step limit is therefore given by Δ

*t*

_{GW}= 1/

*N*. For our standard GWP test case with an isothermal background of

*T*= 300 K and

*N*= 0.018 s

^{−1}we obtain a time step limitation due to gravity wave oscillations of Δ

*t*

_{GW}= 55 s. The time step to be used by the scheme is the minimum of the time steps dictated by the various time step restrictions and a Δ

*t*

_{max}to avoid too-large numerical errors.

#### 2) Low-storage Runge–Kutta and divergence constraint

*m*. The equation for the tendency and the predictor of the momentum are given by

*u*denotes a velocity component and

_{i}*ρ*

^{m}^{+1}is known from the update of the effective continuity equation;

*F*comprises all fluxes, forces, and sources except for the pressure gradient; and

^{M}*α*,

^{m}*β*are the weights for the low-storage Runge–Kutta method by Williamson (1980). The old pressure

^{m}*π*′

^{m}does not guarantee the divergence constraint on

**u***. Therefore, a new pressure

*π*′

^{m+1}is sought that ensures that the velocity

*δ*/

*δx*we obtain for the corrector procedure

_{i}*π*′ with the right-hand side given as the residual divergence of the predicted velocity field

**u*** weighted with

*u** must also be corrected because it is used in the following Runge–Kutta stage; that is, we have to supplement the equation

It is known that a fractional step method deteriorates the temporal convergence order of the overall numerical scheme. To quantify this error we tested three third-order Runge–Kutta schemes with the 1D gravity wave packet test case: the low-storage Runge–Kutta by Williamson (1980) (LS-Will-RK3), a low-storage total-variation-diminishing (TVD) Runge–Kutta (LS-TVD-RK3), and a classical, nonlow-storage TVD Runge–Kutta scheme (CL-TVD-RK3) (Gottlieb and Shu 1998). The results are presented in Fig. 8. If the projection step is switched off all third-order schemes show the expected third-order convergence as Δ*t* → 0. If the projection is switched on the convergence drops down to first order as a simple forward Euler. Note, however, that the overall error of the third-order schemes is one order of magnitude smaller so that it pays off to work with the higher-order schemes.

Convergence study for various time schemes (left) with projection step switched off and (right) with projecting the velocity to satisfy the pseudo-incompressible divergence constraint obtained with the 1D gravity wave packet test case.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Convergence study for various time schemes (left) with projection step switched off and (right) with projecting the velocity to satisfy the pseudo-incompressible divergence constraint obtained with the 1D gravity wave packet test case.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Convergence study for various time schemes (left) with projection step switched off and (right) with projecting the velocity to satisfy the pseudo-incompressible divergence constraint obtained with the 1D gravity wave packet test case.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

### c. ALDM with background stratification

*ρ*′ and apply the reconstruction only to

*ρ*′. The ILES damping term in the flux will only depend on

*t*= 20 min (see section 4 for the setup). On the left the discretization with the total density leads to oscillations in the solution, while the introduction of density fluctuation (right) produces a smooth solution. The small-scale structures on the left are on the grid scale and are numerical artifacts. Especially the oscillations at the top of the domain are unphysical since the solution should remain smooth away from the bubble-induced vortices. The smooth solution on the right compares well with other simulations of the test case in the literature (e.g., Robert 1993; Klein 2009; Mendez-Nunez and Carroll 1994).

Isolines of potential temperature for the hot bubble test case at *t* = 20 min. (left) Severe oscillations if total density *ρ*′.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of potential temperature for the hot bubble test case at *t* = 20 min. (left) Severe oscillations if total density *ρ*′.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of potential temperature for the hot bubble test case at *t* = 20 min. (left) Severe oscillations if total density *ρ*′.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

A similar positive effect can be observed for the 1D gravity wave packet (see section 4 for the setup) shown in Fig. 10. If the total density is reconstructed, severe oscillations appear in the solution (left). ALDM applied only to the density fluctuations leads to a smooth solution (right), which is the correct solution since the gravity wave should not break and produce turbulence in this regime.

Potential temperature of 1D gravity wave packet at *t* = 150 min. (left) Severe oscillations if total density *ρ*′.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Potential temperature of 1D gravity wave packet at *t* = 150 min. (left) Severe oscillations if total density *ρ*′.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Potential temperature of 1D gravity wave packet at *t* = 150 min. (left) Severe oscillations if total density *ρ*′.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

### d. Solving the Poisson problem

Throughout our calculations we use the biconjugate gradient stabilized method (BiCGSTAB) (Meister 1999) algorithm to solve the Poisson problem. A small parameter *ɛ* is introduced in the abort criterion, which will be discussed in section 2.

#### 1) Scaling of the Poisson equation

If we solve the Poisson equation to satisfy *z*_{max} = 150 km with a Poisson-solver tolerance of *ɛ* = 10^{−7} and a fixed time step of 1 s. One way out of this would be to tighten the tolerance—a very inefficient way. We consider a different approach: scaling of the Poisson equation. To see the necessity we analyze the problem by looking at the influence of the divergence error on the effective continuity equation and, consequently, via the momentum equation on the velocity field.

Fluctuation of (left) vertical velocity and (right) potential temperature in a uniform, isothermal atmosphere at rest due to the divergence error after a few minutes’ physical time. Poisson equation without scaling solved with a tolerance of *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Fluctuation of (left) vertical velocity and (right) potential temperature in a uniform, isothermal atmosphere at rest due to the divergence error after a few minutes’ physical time. Poisson equation without scaling solved with a tolerance of *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Fluctuation of (left) vertical velocity and (right) potential temperature in a uniform, isothermal atmosphere at rest due to the divergence error after a few minutes’ physical time. Poisson equation without scaling solved with a tolerance of *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

*ρ*+

_{t}**∇**· (

*ρ*

**u**) = 0 with the density split as

*ɛ*and using

*w*′ created because of the divergence error is height dependent and will grow exponentially like

*w*′ we see from (54) that the divergence error in the density should scale as

*ɛ*= 10

^{−7}we obtain the results presented in Fig. 12. With the same computational effort, velocity and potential temperature perturbations were reduced by four orders of magnitude. Note for the anelastic divergence constraint a scaling with

Fluctuation of (left) vertical velocity and (right) potential temperature in a uniform, isothermal atmosphere at rest due to the divergence error. Poisson equation with * * scaling solved with a tolerance of *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Fluctuation of (left) vertical velocity and (right) potential temperature in a uniform, isothermal atmosphere at rest due to the divergence error. Poisson equation with * * scaling solved with a tolerance of *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Fluctuation of (left) vertical velocity and (right) potential temperature in a uniform, isothermal atmosphere at rest due to the divergence error. Poisson equation with * * scaling solved with a tolerance of *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

#### 2) Physical convergence criterion

Smolarkiewicz et al. (1997) discussed stopping criteria for the iterative solution of the Poisson problem in the context of the Boussinesq system. Their main argument is a physical one: an iteration should be stopped if the relative density change Δ*ρ*/*ρ* due to the divergence error within one time step Δ*t* is smaller than some *ɛ*. A convergence to machine precision would be a waste of time since other discretization errors are then orders of magnitudes larger.

#### 3) Correcting the divergence error

Smolarkiewicz and Margolin (1994) showed that there can be an enormous difference between the Eulerian and the semi-Lagrangian approach: the flux-form formulation needs a much smaller Poisson-solver tolerance *ɛ* in the iterative elliptic solver to avoid unphysical solutions than the advective formulation.^{3} This leads to a higher number of iterations and increase in computing time. In our model we also observed this inefficiency of the Poisson solver and propose the following remedy, which, on the one hand, violates the conservation property of the effective continuity equation. On the other hand this error is of the order of the Poisson-solver tolerance and is therefore acceptable—depending on the application.

*ρ*+ ∇ · (

_{t}*ρ*

**u**) = 0 in a different form:

*ɛ*is very small the term can be neglected in all other cases it has an (unphysical) effect. To overcome this discrepancy, we simply subtract this term from the equation, which is the same as adding it as a source term in the effective continuity equation:

*ɛ*= 10

^{−5}(top) the scheme is not even stable and for

*ɛ*= 10

^{−7}(bottom) there are oscillations in the higher-altitude regions. In contrast, the introduction of the source term in the effective continuity equation stabilizes the scheme and produces correct results for tolerances as low as

*ɛ*= 10

^{−5}. In Table 1 we see how the average number of iterations per call to BiCGSTAB

^{4}is reduced by introducing the divergence-error correction. The reason for this behavior is clear: if the error is not produced in the effective continuity equation, no unphysical dynamic is induced in the momentum equation—and an initially unperturbed, divergence-constraint satisfying state maintains this divergence constraint. Note that for

*ɛ*≤ 10

^{−8}the divergence-error correction no longer has influence. Also note that in some instances BiCGSTAB is not called at all, leading to average numbers smaller than 1.

Potential temperature for the 1D gravity wave packet (see section 4) (left) without divergence-error correction and (right) with the divergence-error correction as given in (64) for a Poisson-solver tolerance of (top) *ɛ* = 10^{−5} and (bottom) *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Potential temperature for the 1D gravity wave packet (see section 4) (left) without divergence-error correction and (right) with the divergence-error correction as given in (64) for a Poisson-solver tolerance of (top) *ɛ* = 10^{−5} and (bottom) *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Potential temperature for the 1D gravity wave packet (see section 4) (left) without divergence-error correction and (right) with the divergence-error correction as given in (64) for a Poisson-solver tolerance of (top) *ɛ* = 10^{−5} and (bottom) *ɛ* = 10^{−7}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Average number of iterations of the Poisson solver (BiCGSTAB) for the 1D GWP test case; refer to section 4.

## 4. Test case results

### a. Hot bubble test case

*θ*

_{0}= 300 K. The initial perturbation by the hot thermal is given by

*r*

_{0}= 2.5 km and the initial height is

*z*= 2.75 km. The bubble is placed horizontally in the middle of the domain. This test case was chosen to demonstrate the importance of transporting only the density perturbation instead of the full density; see Fig. 9 and the discussion in section 3c. The test shows interfacial instabilities as already reported in Grabowski and Clark (1991) for a thermal with moist physics and a high-resolution (direct simulation) approach.

_{c}*θ*

_{tr}= 300 K, the tropopause is set at

*z*

_{tr}= 12 km, and the stratosphere is assumed isothermal with a constant temperature given by the temperature at the tropopause

*z*=

*z*

_{tr}. Nevertheless, the updraft produced by the hot bubble induces perturbations at the tropopause, which travel as gravity waves into the stratosphere. Wave crests and troughs are visible in (14) as elliptic isolines with positive and negative values of potential temperature.

Isolines of potential temperature perturbation *z* > 12 km after 20 min. The isolines range from −3.2 to 2.2 K in steps of 0.5 K. Positive perturbations are marked with a “+” and negative perturbations with a “−”.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of potential temperature perturbation *z* > 12 km after 20 min. The isolines range from −3.2 to 2.2 K in steps of 0.5 K. Positive perturbations are marked with a “+” and negative perturbations with a “−”.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of potential temperature perturbation *z* > 12 km after 20 min. The isolines range from −3.2 to 2.2 K in steps of 0.5 K. Positive perturbations are marked with a “+” and negative perturbations with a “−”.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

### b. Bubble test case by Robert

*θ*

_{0}= 0.5 K,

*r*

_{0}= 150 m, and

*σ*= 50 m, and is positioned at

*x*

_{0}= 500 m,

*z*

_{0}= 300 m. The smaller cold bubble has the data Δ

*θ*

_{0}= −0.15 K,

*r*

_{0}= 0 m, and

*σ*= 50 m, and is positioned at

*x*

_{0}= 560 m,

*z*

_{0}= 640 m. The domain spans [−500 m, 500 m] × [0, 1000 m]. No-slip boundary conditions are applied at all four walls of the domain. The CFL number is set to 0.9, leading to a time step of 3.0 s for 100 × 100 and 1.5 s for the 200 × 200 resolution. An interesting aspect is the existence of two scales introduced by the large and small bubble and the fact that the cold bubble is horizontally “off centered,” which creates a highly unsymmetric solution. In Fig. 15 we compare the isolines of density perturbation

Isolines of density perturbation *ρ*′ = −2.2 to *ρ*′ = 0.5 g m^{−3} in steps of Δ*ρ*′ = 0.3 g m^{−3}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of density perturbation *ρ*′ = −2.2 to *ρ*′ = 0.5 g m^{−3} in steps of Δ*ρ*′ = 0.3 g m^{−3}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of density perturbation *ρ*′ = −2.2 to *ρ*′ = 0.5 g m^{−3} in steps of Δ*ρ*′ = 0.3 g m^{−3}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

A variant of Robert’s test case with a uniform horizontal background wind *u*_{0} = 1.67 m s^{−1} is shown in Fig. 16 with a resolution of 200 × 200. The velocity is set so that the bubbles fully traverse the periodic domain within 10 min. The plotted levels for the isolines of density perturbation

Isolines of density perturbation for Robert’s test case (left) without initial wind and (right) with a uniform mean wind *u*_{0} = 1.67 m s^{−1}. (top center) Initial setting (second row) after (*ρ*′ = −2.2 to *ρ*′ = 0.5 g m^{−3} in steps of Δ*ρ*′ = 0.3 g m^{−3}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of density perturbation for Robert’s test case (left) without initial wind and (right) with a uniform mean wind *u*_{0} = 1.67 m s^{−1}. (top center) Initial setting (second row) after (*ρ*′ = −2.2 to *ρ*′ = 0.5 g m^{−3} in steps of Δ*ρ*′ = 0.3 g m^{−3}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of density perturbation for Robert’s test case (left) without initial wind and (right) with a uniform mean wind *u*_{0} = 1.67 m s^{−1}. (top center) Initial setting (second row) after (*ρ*′ = −2.2 to *ρ*′ = 0.5 g m^{−3} in steps of Δ*ρ*′ = 0.3 g m^{−3}.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

### c. 1D gravity wave packet

*T*

_{00}= 300 K a wave packet is superimposed with the initial buoyancy amplitude

*m*

_{0}= 2

*π*/

*λ*

_{z}_{,0}and the normalized amplitude

*a*= 0.1, the isothermal Brunt–Väisälä frequency for

*T*

_{00}= 300 K of

*N*= 0.018 s

^{−1}, the half-width

*σ*= 5 km, and the center of the wave packet at

*z*= 30 km. Using the polarization relations we set the following initial fields:

_{c}*λ*=

_{x}*λ*= 1 km, defining the initial wavenumber

_{z}**k**

_{0}= (

*k*

_{0},

*m*

_{0}) and the intrinsic frequency

*l*,

_{x}*l*) = (1 km, 60 km) at a resolution of

_{z}*n*×

_{x}*n*= 64 × 3840. In both test cases the domain is periodic in the horizontal and has a no-slip solid wall boundary condition at the bottom and top. A sponge layer at the top can be switched on to avoid spurious reflections of the gravity wave packet.

_{z}In Rieper et al. (2012, manuscript submitted to *J. Fluid Mech.*) the extended, weakly nonlinear Wentzel–Kramers–Brillouin (WKB) theory by Achatz et al. (2010) is verified—that is, a WKB model and a full model^{5} are compared with respect to the predictions of the WKB theory. Here we recall the major results in short: there is very good agreement for the propagation of wave 1 (initial wavenumber **k**_{0}). The full model shows a wave 2 (wavenumber 2**k**_{0}) oscillating about a smoothly increasing amplitude predicted by the WKB theory. For smaller *λ* (i.e., for smaller *ɛ* in the multiscale asymptotic ansatz), these oscillations vanish. Higher harmonics should not be induced within the orders of magnitude considered in the WKB theory (Achatz et al. 2010) and the full model only produces higher harmonics of a distinctly smaller order of magnitude.

In this text we analyze the behavior of full and WKB model with respect to numerical issues: influence of resolution, flux function, Poisson-solver tolerance, and the divergence-error correction introduced in section 3. Throughout these calculations we used a fixed (small) time step Δ*t* = 1.0 s in order to fully resolve the buoyancy oscillations. The corresponding CFL numbers depend on the chosen resolution and are stated at due place.

In Fig. 17 it is shown how the various harmonics evolve in time in the full model and according to the WKB theory depending on the spatial resolution. Shown are the amplitude maxima of buoyancy for the wavenumbers *α***k**_{0} with *α* = 2, 3, 4, and 5. The resolution *n _{λ}* is given with respect to the number of points per wavelength of the initial wave with

**k**

_{0}. The time step of 1 s leads to a CFL number of

*n*= 16—corresponding to a domain resolution of

_{λ}*n*×

_{x}*n*= 16 × 960—the wave 2 amplitude falls behind the WKB-predicted value. On the other hand, the higher harmonics (

_{z}*α*> 2), which should be absent according to WKB, are of comparable magnitude as wave 2. Obviously, a poor resolution leads to a feed in of energy from wave 2 to the higher harmonics. If the resolution is doubled (right of Fig. 17) the higher harmonics are clearly of a smaller order of magnitude. If the resolution is doubled again (not shown here) the higher harmonics practically vanish and the WKB predictions are well met. Note that the wave 1 amplitude of the full model (also not shown) always agrees very well with the WKB prediction.

Amplitude maximum of buoyancy as a function of time for the 1D GWP for wavenumber 2 to 5 calculated with the full model at a resolution of *n _{λ}* = 16 points per wavelength for the initial wave with (left)

**k**

_{0}, (right)

*n*= 32, and the WKB model (solid line).

_{λ}Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Amplitude maximum of buoyancy as a function of time for the 1D GWP for wavenumber 2 to 5 calculated with the full model at a resolution of *n _{λ}* = 16 points per wavelength for the initial wave with (left)

**k**

_{0}, (right)

*n*= 32, and the WKB model (solid line).

_{λ}Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Amplitude maximum of buoyancy as a function of time for the 1D GWP for wavenumber 2 to 5 calculated with the full model at a resolution of *n _{λ}* = 16 points per wavelength for the initial wave with (left)

**k**

_{0}, (right)

*n*= 32, and the WKB model (solid line).

_{λ}Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

At the resolution *n _{λ}* = 32 the influence of the flux function (not shown here) is only marginal. We compare three different flux functions: central difference scheme (CDS), MUSCL, and ALDM produce a second harmonic of comparable magnitude. This amplitude is smoothly varying for MUSCL and ALDM but shows a rough evolution for CDS. The differences appear in the higher harmonics. CDS practically does not produce any higher harmonics. MUSCL has very small oscillations in the odd harmonics of

*O*(1%) of the second harmonic, while ALDM has

*O*(10%)—oscillations in the third harmonic, and

*O*(1%)—oscillations in the harmonics higher than 3 for a Poisson-solver tolerance of

*ɛ*= 10

^{−9}.

The tolerance of the Poisson solver also influences the spectrum of the solution as can be seen in Fig. 18. At a tolerance of *ɛ* = 10^{−5} the full model produces spurious amplitudes in the higher harmonics: *α* > 1. Only for small tolerances *ɛ* < 10^{−6} do these harmonics disappear. Note that wave 1 (not shown here) is in good agreement to the WKB predictions for tolerances as large as *ɛ* = 10^{−5}. These calculations were done at a high resolution of *n _{λ}* = 64 with a fixed time step of 1 s and a corresponding CFL number of

Amplitude maximum of buoyancy as a function of time for the 1D GWP for wavenumber 2 to 5 calculated with the full model at a Poisson-solver tolerance of (left) *ɛ* = 10^{−5}, (right) *ɛ* = 10^{−7}, and the WKB model (solid lines).

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Amplitude maximum of buoyancy as a function of time for the 1D GWP for wavenumber 2 to 5 calculated with the full model at a Poisson-solver tolerance of (left) *ɛ* = 10^{−5}, (right) *ɛ* = 10^{−7}, and the WKB model (solid lines).

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Amplitude maximum of buoyancy as a function of time for the 1D GWP for wavenumber 2 to 5 calculated with the full model at a Poisson-solver tolerance of (left) *ɛ* = 10^{−5}, (right) *ɛ* = 10^{−7}, and the WKB model (solid lines).

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

The tolerance of the Poisson solver can be relaxed if the divergence-error correction as described in section 3 is applied. If it is switched off we obtain the result shown in Fig. 19. The higher harmonics do not agree with WKB predictions in contrast to the results shown on the right side of Fig. 18 with the divergence-error correction switched on. Obviously, even wave 1 (left) is completely wrong; see Fig. 19 left. These calculations were also done with *n _{λ}* = 64, a fixed time step of 1 s, and a corresponding CFL number of

Amplitude maximum of buoyancy as a function of time for the 1D GWP for (left) wavenumber 1 and for (right) the higher harmonics 2 to 5 calculated with the full model at a Poisson-solver tolerance of *ɛ* = 10^{−7} without the divergence-error correction. The solid line shows the WKB-predicted behavior.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Amplitude maximum of buoyancy as a function of time for the 1D GWP for (left) wavenumber 1 and for (right) the higher harmonics 2 to 5 calculated with the full model at a Poisson-solver tolerance of *ɛ* = 10^{−7} without the divergence-error correction. The solid line shows the WKB-predicted behavior.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Amplitude maximum of buoyancy as a function of time for the 1D GWP for (left) wavenumber 1 and for (right) the higher harmonics 2 to 5 calculated with the full model at a Poisson-solver tolerance of *ɛ* = 10^{−7} without the divergence-error correction. The solid line shows the WKB-predicted behavior.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

### d. Breaking of a 2D gravity wave packet

*λ*=

_{x}*λ*= 4 km and a normalized amplitude factor of

_{z}*a*= 0.9. The domain is 80 km × 80 km and resolved with

*n*×

_{x}*n*= 640 × 640. The time step was fixed to 1 s leading to a CFL number of

_{z}*T*

_{00}= 300 K. In Fig. 20 the isolines of the total potential temperature

Isolines of the total potential temperature

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of the total potential temperature

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Isolines of the total potential temperature

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

## 5. Summary and conclusions

We have presented a conservative way of discretizing Durran’s pseudo-incompressible equations, which was—to the best of our knowledge—until now only done by Smolarkiewicz and Dörnbrack (2008) and by Smolarkiewicz and Szmelter (2011) on unstructured meshes. New is the implementation of a turbulence model right into the numerical flux function. The adaptive local deconvolution method (ALDM) by S. Hickel (2011, unpublished manuscript), Hickel et al. (2006, 2007), and Hickel and Larsson (2008) is for the first time used in the context of non-Boussinesq flows with a stratified background.

We analyze numerical difficulties and propose ways around them:

To avoid spurious oscillations and to avoid the activation of the turbulence model on an unperturbed atmosphere, ALDM must only be applied to the deviation

from the background. The physically motivated abort criterion proposed by Smolarkiewicz et al. (1997) for solving the elliptic Helmholtz equation in the context of the Euler equation is adapted to the pseudo-incompressible system.

To avoid oscillations in a uniform atmosphere at high altitude the Poisson equation has to be scaled with

. This causes a height-independent divergence error in the velocity field, allowing us to reduce the Poisson-solver tolerance to acceptable values with respect to the overall computational effort. A correction term in the effective continuity equation allows us to reduce the number of iterations of the Poisson solver even more. It reduces the effect of the residual divergence error present in the velocity field after the projection step.

With the hot–cold bubble test case by Robert (1993) we validate the code for convective problems. The similarity between the solutions shows three things:

Euler and Durran’s equation produce similar results if applied to convective problems.

Upwind schemes and ALDM produce similar results if applied to well-resolved laminar flows and ALDM has less numerical dissipation on coarse grids.

The solution only weakly depends on a uniform mean background wind.

With the hot bubble test case by Mendez-Nunez and Carroll (1994) we demonstrate the relevance of applying ALDM only to the density perturbations and the capability of the model to cope with convection in the presence of a stratified background.

The 1D gravity wave packet (GWP) test case is used to demonstrate the influence of the numerics on the spectral development of a GWP. We compare the results with those predicted by the extended WKB theory (Achatz et al. 2010):

If the resolution is too low, higher harmonics are excited—which is not of physical but only of numerical origin.

The tolerance of the Poisson solver has a direct impact on the spectrum, especially on the higher harmonics.

The introduction of a divergence-error correction improves the quality of the spectrum for lower tolerance of the Poisson solver.

In a 2D GWP breaking test case we visualize how the ALDM turbulence model keeps a smooth solution without creating unphysical oscillations on the grid scale. Since this text focuses on numerical issues we do not go further into analyzing the turbulence characteristics, which is planned for future work.

As turbulence in real GW breaking events is a three-dimensional phenomenon, it remains open to show how a 3D GWP breaks and how the ALDM scheme behaves in such a more physical 3D context. Since calculations in 3D are time consuming, this part of the research has to be postponed until a parallel version of pincFloit is available.

So far the model is restricted to the dry atmosphere. We plan to include moist processes in order to be able to study the interaction between gravity waves and clouds. With the implicit turbulence scheme the moist quantities would be transported as a passive tracer—an application for which ALDM has already been tested (Hickel et al. 2007). The latent heat term appears on the right-hand side of the divergence constraint.

## Acknowledgments

We thank the Leibniz-Gemeinschaft (WGL) for support within their PAKT program. S.H. and U.A. thank Deutsche Forschungsgemeinschaft for partial support through the MetStröm Priority Research Program (SPP 1276) and through Grants HI 1273/1-1 Ac71/4-1.

## APPENDIX A

### Momentum Flux Functions and Corresponding Density Interpolations

#### a. Momentum fluxes

#### b. Density interpolation

## APPENDIX B

### Reconstruction Procedure

^{B1}value

*i*at the right cell face (

*λ*= 1/2), with a quadratic polynomial (

*k*= 3) with the left-most stencil (

*r*= 2) consisting of

*l*= 0, 1, 2. In general the weights of the filtered values are given by (Hickel et al. 2006):

#### a. Constant reconstruction: k *= 1*

*k*= 1 we have reconstruction with a constant function as depicted in Fig. B1. The indices can assume the following values:

*x*

_{i}_{−1/2},

*x*

_{i}_{+0}, and

*x*

_{i}_{+1/2}are given by

Reconstruction of *φ* in cell *i*.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Reconstruction of *φ* in cell *i*.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Reconstruction of *φ* in cell *i*.

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

#### b. Linear reconstruction: k *= 2*

*k*= 2 the remaining coefficients take the values

Two possible reconstructions of *φ* in cell *i* with linear functions with weights *α* for corresponding filtered values

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Two possible reconstructions of *φ* in cell *i* with linear functions with weights *α* for corresponding filtered values

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

Two possible reconstructions of *φ* in cell *i* with linear functions with weights *α* for corresponding filtered values

Citation: Monthly Weather Review 141, 3; 10.1175/MWR-D-12-00026.1

#### c. Quadratic reconstruction: k *= 3*

The quadratic reconstruction is depicted in Fig. 3. Note that the obtained functions do not interpolate the filtered values *φ* and not to the function itself. The weights for the filtered values are given in the following tableau for *λ* = ±1/2:

#### d. Blending of first, second, and third-order interpolants

In essentially nonoscillatory (ENO) schemes, a definite polynomial degree *k* for the interpolant functions is chosen that determines the spatial order of the scheme. A stencil-selection algorithm then chooses a stencil *r* in such a way as to avoid reconstruction across discontinuities. Contrary to this, weighted essentially nonoscillatory (WENO) schemes use all stencils *r* = 0, …, *k* − 1 of degree *k*, where the interpolants across discontinuities receive smaller weights than smooth regions, thus keeping oscillations low.

*k*= 1, 2, to 3. Thus, we sum over all stencils

*r*= 0, …,

*k*− 1 and all polynomial degrees

*k*= 1, 2, 3 to obtain the reconstructed variables at cell faces

*K*− 1, where

*K*is the degree of polynomials used. In ALDM the weights are tuned to represent subgrid-scale turbulence via the truncation error of the various orders of reconstruction. They are defined like

*γ*’s are tuning parameters and the

*β*’s are smoothness indicators given by

*ɛ*= 10

_{β}^{−10}avoids division by 0 in smooth or nonvarying regions. We calculate the smoothness indicator for the various instances of

*k*and

*r*.

#### e. Properties of the *γ*s

The properties of the 18 parameters *γ* are given in Hickel (2008) leading to a reduction of the numbers of free parameters down to 4, named here *c*_{1} to *c*_{4}. We summarize

*σ*is a parameter appearing in the definition of the numerical flux function as defined in section 2c.

#### f. Reconstruction weights

##### 1) Constant reconstruction: *ω*_{1,0}

*k*= 1 and

*r*= 0 we only have one admissible stencil, which leads directly to

##### 2) Linear reconstruction: *ω*_{2,0} and *ω*_{2,1}

*γ*’s.

##### 3) Quadratic reconstruction: *β*_{3,0}, *β*_{3,1}, and *β*_{3,2}

*β*

_{3,r}are given by

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^{1}

Please note that in the present paper the Froude number is defined as follows: *u*_{ref} and reference length *l*_{ref}.

^{2}

In the following we only use the term “density” but always refer to the pseudo-incompressible (effective) density.

^{3}

It is also shown that this problem can be circumvented by transporting only the perturbation of the potential temperature instead of the full quantity.

^{4}

Note that each iteration within BiCGSTAB has two calls of the linear operator.

^{5}

Since the flow remains laminar in the analyzed test case in that paper no turbulence model was needed and—without loss of generality—a MUSCL scheme was used.

^{B1}

Note that the bar symbolizes cell-averaged or filtered value in the appendices and not atmospheric background value as in the main text.