1. Introduction
Gravity waves are a common phenomenon in any stably stratified fluid, such as found in the atmosphere of Earth. They can be excited by flow over orography (e.g., Smith 1979; McFarlane 1987), by convection (e.g., Chun et al. 2001; Grimsdell et al. 2010), and by spontaneous imbalance of the mean flow in the troposphere (O’Sullivan and Dunkerton 1995; Plougonven and Snyder 2007). Gravity waves transport energy and momentum from the region where they are forced to the region where they are dissipated (e.g., through breaking), possibly far away from the source region. Various phenomena, such as the cold summer mesopause (Hines 1965) and the quasi-biennial oscillation in the equatorial stratosphere (e.g., Baldwin et al. 2001), cannot be explained nor reproduced in weather and climate simulations without accounting for the effect of gravity waves. See Fritts and Alexander (2003) for an overview of gravity waves in the middle atmosphere. Prusa et al. (1996) found in numerical experiments that (because of wave dispersion) gravity waves generated in the troposphere at a broad wavelength spectrum reach the upper mesosphere as an almost monochromatic wave packet with a horizontal wavelength between a few kilometers and more than 100 km, depending on the horizontal scale of the forcing and the background conditions.
Since most gravity waves have a wavelength that is not well resolved in general circulation models, the effect of gravity waves on the global circulation is usually accounted for by parameterizations based on combinations of linear wave theory (Lindzen 1981), empirical observations of time-mean energy spectra (e.g., Hines 1997), and simplified treatments of the breaking process. See Kim et al. (2003) and McLandress (1998) for reviews of the various standard parameterization schemes.
A common weakness of most parameterization schemes is the oversimplified treatment of the wave breaking process. Improving this point requires a deeper insight into the breaking process that involves generation of small-scale flow features through wave–wave interactions and through wave–turbulence interactions. Since the gravity wave wavelength and the turbulence that eventually leads to energy dissipation into heat span a wide range of spatial and temporal scales, the breaking process is challenging both for observations and numerical simulations. Direct numerical simulations (DNSs) must cover the breaking wave with a wavelength of a few kilometers as well as the smallest turbulence scales (the Kolmogorov length η). The Kolmogorov length depends on the kinetic energy dissipation and the kinematic viscosity. It is on the order of millimeters in the troposphere (Vallis 2006) and approximately 1 m at 80-km altitude (Remmler et al. 2013).
The necessity of resolving the Kolmogorov scale can be circumvented by applying the approach of a large-eddy simulation (LES): that is, by parameterizing the effect of unresolved small eddies on the resolved large-scale flow. This can be necessary in cases where DNS would be too expensive [e.g., in investigating the dependence of the gravity wave breaking on several parameters (propagation angle, wavelength, amplitude, viscosity, and stratification) at the same time]; for problems in which many wavelengths need to be resolved, such as propagation of a wave packet or wave train through a variable background (Lund and Fritts 2012) or modeling realistic cases of waves generated by topography or convection; for validating quasi-linear wave-propagation theory (Muraschko et al. 2014); or for validating gravity wave–drag parameterization schemes.
The subgrid-scale parameterization of turbulence is, of course, a source of uncertainty and, where possible, should be validated against fully resolved DNSs or observations for every type of flow for which it is to be used. Many numerical studies of breaking gravity waves rely on the LES principle without such a validation (e.g., Winters and D’Asaro 1994; Lelong and Dunkerton 1998a,b; Andreassen et al. 1998; Dörnbrack 1998; Afanasyev and Peltier 2001).
Recent studies (Fritts et al. 2009a,b, 2013; Fritts and Wang 2013) have presented highly resolved, high Reynolds number DNSs of a monochromatic gravity wave breaking. However, they do not take into account the Coriolis force, which has a large influence on the dynamics of breaking for low-frequency gravity waves (Dunkerton 1997; Achatz and Schmitz 2006b), often referred to as inertia–gravity waves (IGWs), as opposed to high-frequency gravity waves (HGWs). The Coriolis force induces an elliptically polarized transverse velocity field in IGWs, and the velocity component normal to the plane of propagation of the wave has its maximum shear at the level of minimum static stability. Dunkerton (1997) and Achatz and Schmitz (2006b) showed that this strongly influences the orientation of the most unstable perturbations.
An important aspect in setting up a simulation of a gravity wave breaking event is the proper choice of the domain size and initial conditions. While the gravity wave itself depends on one spatial coordinate and has a natural length scale given by its wavelength, the breaking process and the resulting turbulence are three-dimensional, and proper choices have to be made for the domain sizes in the two directions perpendicular to the wave vector. Achatz (2005) and Achatz and Schmitz (2006a) analyzed the primary instabilities of monochromatic gravity waves of various amplitudes and propagation directions using normal-mode and singular-vector analysis, and Fruman and Achatz (2012) extended this analysis for IGWs by computing the leading secondary singular vectors with respect to a time-dependent simulation of the perturbed wave. (Normal-mode analysis is not suited to time-dependent basic states, while singular-vector analysis, whereby the perturbations for which energy grows by the largest factor in a given optimization time, is always possible.) They found that the wavelength of the optimal secondary perturbation can be much shorter than the wavelength of the original wave. Thus, the computational domain for a three-dimensional simulation need not necessarily have the size of the base wavelength in all three directions. They proposed the following multistep approach to set up the domain and initial conditions for a DNS of a given monochromatic gravity wave:
- solution (in the form of normal modes or singular vectors) of the governing (Boussinesq) equations linearized about the basic-state wave, determining the primary instability structures;
- two-dimensional (in space) numerical solution of the full nonlinear equations using the result of stage 1 as initial condition;
- solution in the form of singular vectors (varying in the remaining spatial direction) of the governing equations linearized about the time-dependent result of stage 2; and
- three-dimensional DNS using the linear solutions from stages 1 and 3 as initial conditions and their wavelengths for the size of the computational domain.
Having these properly designed DNS results available, we can now use them for the validation of computationally less expensive methods. Hence, the present study analyzes the suitability of different LES methods for the cases presented by Fruman et al. (2014): namely, an unstable IGW, a stable IGW, and an unstable HGW, all of them with a base wavelength of 3 km. The first LES method to be applied is the adaptive local deconvolution method (ALDM) of Hickel et al. (2006, 2014). It is an “implicit” LES method, since the subgrid-scale (SGS) stress parameterization is implied in the numerical discretization scheme. Based on ALDM for incompressible flows and its extension to passive scalar mixing Hickel et al. (2008), Remmler and Hickel (2012, 2013), and Rieper et al. (2013) successfully applied ALDM to stably stratified turbulent flows. For the present study, the numerical flux function for the active scalar in ALDM has been modified to prevent the method from generating spurious oscillations in partially laminar flow fields.
The second method to be applied is the well-known Smagorinsky (1963) method with the dynamic estimation of the spatially nonuniform model parameter proposed by Germano et al. (1991) and refined by Lilly (1992). The third LES method is a “naïve” approach with a simple central discretization scheme and no explicit SGS parameterization. This method is theoretically dissipation free but can lead to numerical instability if the turbulence level is high (which was the reason for the development of the first explicit SGS parameterization by Smagorinsky 1963). However, the method is computationally inexpensive and can be used in some cases without problems, as we will show.
We apply these methods to the three gravity wave test cases using grids of different refinement levels with the goal of using as few grid cells as possible while still obtaining good agreement with the DNS results. We also run small ensembles for each simulation with only slightly different initial conditions to get an estimate of the sensitivity and variability of the results. All this is done in a three-dimensional domain (with the same domain size as the DNS) and in a two-dimensional domain in which the two dimensions are chosen to be parallel to the wave vectors of the gravity wave and of the most important growing primary perturbation [without the addition of the secondary singular vector (cf. step 2 above)]. Because the velocity and vorticity fields are three-dimensional and because the turbulent cascade is direct (energy moves to smaller length scales), these simulations are sometimes called 2.5D. Fruman et al. (2014) found that 2.5D and 3D results are broadly very similar for the inertia–gravity wave test cases considered here.
The paper is organized as follows. In section 2, the governing equations used for the simulations are presented along with properties of the inertia–gravity wave solutions and the energetics of the system. Section 3 describes the numerical methods used, in particular the three LES schemes. The three test cases are reviewed in section 4, and the results of the simulations are discussed in sections 5–7.
2. Governing equations
Assuming the vertical wavelength of the inertia–gravity wave is small compared to the density scale height of the atmosphere, the dynamics are reasonably well approximated by the Boussinesq equations on an f plane. For mathematical convenience, we further assume that the molecular viscosity and diffusion, as well as the Brunt–Väisälä frequency of the background, are constants, independent of space and time.
Rotated coordinate system and computational domain (gray box) for the monochromatic gravity wave (after Remmler et al. 2013).
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
3. Numerical methods
a. The INCA model
With our flow solver [solver for the (in)compressible Navier–Stokes equations on Cartesian adaptive grids (INCA)], the Boussinesq equations are discretized by a fractional step method on a staggered Cartesian mesh. The code offers different discretization schemes depending on the application, two of which are described below. For time advancement, the explicit third-order Runge–Kutta scheme of Shu (1988) is used. The time step is dynamically adapted to satisfy a Courant–Friedrichs–Lewy condition.
The spatial discretization is a finite-volume method. We use a second-order central difference scheme for the discretization of the diffusive terms and for the pressure Poisson solver. The Poisson equation for the pressure is solved at every Runge–Kutta sub step. The Poisson solver employs the fast Fourier transform in the vertical (i.e., ζ) direction and a stabilized biconjugate gradient (BiCGSTAB) solver (van der Vorst 1992) in the horizontal (
b. The adaptive local deconvolution method
The reconstruction of the unfiltered solution on the represented scales is based on Harten-type deconvolution polynomials. Different polynomials are dynamically weighted depending on the smoothness of the filtered solution. The regularization is obtained through a tailored numerical flux function operating on the reconstructed solution. Both the solution-adaptive polynomial weighting and the numerical flux function involve free model parameters that were calibrated in such a way that the truncation error of the discretized equations optimally represents the SGS stresses of isotropic turbulence (Hickel et al. 2006). This set of parameters was not changed for any subsequent applications of ALDM. For the presented computations, we used an implementation of ALDM with improved computational efficiency (Hickel and Adams 2007).
c. Dynamic Smagorinsky method
d. Central discretization scheme
4. Test cases
According to Prusa et al. (1996), gravity waves arriving at the upper mesosphere tend to be almost monochromatic, with horizontal wavelengths ranging from a few kilometers to more than 100 km, and with vertical wavelengths of a few kilometers. These waves break at altitudes between 65 and 120 km. We investigate three different cases of monochromatic gravity waves in an environment representative of the upper mesosphere at an altitude of approximately 80 km. For the atmospheric parameters, see Table 1, and for the wave parameters, see Table 2. All three waves have a wavelength of 3 km and the wave phase is such that, in the rotated coordinate system, the maximum total buoyancy gradient within the wave is located at 
Atmospheric parameters.
Parameters of the initial conditions for the investigated test cases. 
Initial horizontally (
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
Case 1 is a statically unstable inertia–gravity wave with a wave period of 8 h and a phase speed of 0.1 m s−1. The vertical and horizontal wave lengths are similar to those actually observed by Hoffmann et al. (2005) in wind radar measurements at an altitude of approximately 85 km. The wavelength of the leading transverse normal mode (primary perturbation) is somewhat longer than the base wavelength (
Case 2 is also an inertia–gravity wave with the same period and phase speed as case 1, but with an amplitude below the threshold of static instability. The wave is perturbed by the leading transverse primary singular vector (
Case 3 is a statically unstable high-frequency gravity wave with a period of 15 min and a phase speed of 3.3 m s−1 perturbed with the leading transverse primary normal mode (
The three different cases were chosen to represent a wide range of different configurations of breaking gravity waves. They especially differ in the duration of the breaking compared to the wave period. In case 1, the breaking duration is slightly smaller than the wave period, and the breaking involves multiple bursts of turbulence. In case 2, the breaking lasts only for a short time compared to the wave period, and, in case 3, the breaking lasts longer than one wave period.
5. Case 1: Unstable inertia–gravity wave
a. Three-dimensional DNS
Fruman et al. (2014) showed that, in 2.5D simulations, a small initial random disturbance of the flow field can lead to different global results. To investigate whether the same applies to full 3D simulations of the same case and whether the LES method has an influence on this variability, we added two new DNSs (640 × 64 × 500 cells) to the results of Remmler et al. (2013) to have a very small ensemble of four simulations from which we can compute averages and standard deviations. For these new simulations, a very small amount of white noise was added to all three velocity components at the initial time. In Fig. 3, we show the ensemble average of the amplitudes 
Statically unstable IGW (3D). Base wave amplitudes a and total dissipation rates 
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
For the present case, the wave breaking consists of a series of three single breaking events. Each of those events is characterized by a peak in the energy dissipation and by an enhanced amplitude decrease. The strongest breaking event is initialized by the initial perturbations and starts directly at the beginning of the simulation. It involves overturning and generation of turbulence in the whole computational domain. The intensity of this primary breaking event is very similar in all ensemble members, independent of the resolution and initial white noise. The second breaking event around t ≈ 4 h is preceded by an instability of the large-scale wave and generates only a small amount of turbulence in the unstable half of the domain. The third breaking event around t ≈ 5 h is caused by a small amount of remaining turbulence from the first breaking event, which was generated in the stable part of the wave. At the time of the third breaking event, the wave phase has traveled approximately half a wavelength, so the unstable part of the wave has reached the region of the remaining turbulence at this time. For details of the wave breaking process, we refer to Remmler et al. (2013) and Fruman et al. (2014).
The amplitude variations in the 3D DNSs are very small. The ensemble members diverge slightly during a very weak breaking event at t ≈ 8 h, which has very different intensity in the four simulations. The total dissipation rate varies significantly among the ensemble members during the weak breaking events but not during the first strong breaking event.
b. Three-dimensional LES
We simulated the 3D setup of case 1 using three different LES resolutions, which we refer to as fine (100 × 24 × 80 cells, corresponding to a cell size of 39.8 m × 17.7 m × 37.5 m), medium (24 × 12 × 80 cells), and coarse (24 × 12 × 20 cells). We chose these resolutions after a series of numerical experiments that showed two main results: (i) the horizontal (i.e., in the 
In LES, it is easily affordable to run small ensembles for many different simulations. For all presented 3D LES results, we performed the same simulation eight times with some low-level white noise superposed on the initial condition (consisting of the base wave and its leading primary and secondary perturbations) and once with no added noise. The results of these nine realizations were then averaged. The average amplitudes of simulations with three different resolutions and four different LES methods [standard ALDM with 
Using the fine LES grid, the average wave amplitude is quite well predicted by standard ALDM, DSM, and CDS4. Modified ALDM dissipates slightly too much energy, while standard ALDM shows very large variations between ensemble members.
With the medium grid, the three SGS models (i.e., DSM and the two versions of ALDM) yield good agreement with the DNS, both in the average amplitude and in the variations among ensemble members. Only CDS4 (without an SGS model) creates a bit too much dissipation and far too much variability. The analysis of the total dissipation rates in Fig. 3d shows that the exact evolution of the dissipation is reproduced by none of the LES methods. However, the results with modified ALDM, DSM, and CDS4 are acceptably close to the DNS results. The result from standard ALDM strongly oscillates in time despite being an ensemble average. For clarity of the figure, we did not plot the error bars for this curve, but it is nevertheless obvious that the variation among the ensemble members with standard ALDM is much larger than with the other methods.
With the coarse resolution, ALDM and CDS4 dissipate far too much energy and, hence, predict a too-quick wave decay. The flow physics are not correctly reproduced. Only DSM produces an acceptable result, although the wave amplitude at the end of the breaking is considerably lower than in the DNS. The variability in the DSM results is even smaller than in the DNS, which is not necessarily an indication of a good approximation of the unresolved turbulent scales.
The strong variations of the wave amplitude (fine resolution) and total dissipation rate (medium resolution) in the simulations with standard ALDM motivated the development of the modification described in section 3b. In Fig. 4, we compare Hovmöller diagrams of resolved kinetic energy dissipation (as an indicator of velocity fluctuations) averaged in 
Statically unstable IGW (3D). Hovmöller plots of horizontally averaged (resolved) dissipation of kinetic energy. (a) DNS (640 × 64 × 500 cells) and (b)–(d) LES (100 × 24 × 80 cells). The dashed line indicates a fixed position in space.
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
The first breaking event lasts for about 1 h and is associated with the strongest turbulence. This peak is predicted quite differently by the different SGS models. We show the energy dissipation during the first 2 h in Fig. 5. In the ALDM and DSM simulations, only a minor part of the total energy dissipation is resolved because of the coarse resolution; the remainder is provided by the implicit (
Statically unstable IGW (3D). Comparison of (a),(c),(e) resolved (
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
For the CDS4 simulations, where no SGS parameterization is applied (Fig. 5e), the resolved dissipation 
Statically unstable IGW (3D). Distributions of total, kinetic, and potential energy over vertical wavelength 
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
The energy dissipation can be decomposed into mechanical energy dissipation 
Instantaneous vertical energy spectra averaged in the 
At t = 2 h, the agreement of the 
c. 2.5D simulations
The results of the 2.5D simulations (DNS and LES) are summarized in Fig. 7. Corresponding to the 3D LESs, we chose LES grids with high (100 × 80 cells), medium (24 × 80 cells), and coarse (24 × 20 cells) resolution and performed the same LES eight times with some low-level white noise superposed on the initial condition (consisting of the base wave and its leading primary perturbation) and once with no added noise. The results of these nine realizations were then averaged. The reference 2.5D DNSs were run at a resolution of 660 × 500 cells (see Fruman et al. 2014). An ensemble of eight DNSs was used for the calculation of mean values and standard deviations. For a detailed comparison of 2.5D and 3D DNS results, see Fruman et al. (2014).
Statically unstable IGW (2.5D). Base wave amplitude a and total dissipation rate 
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
The results obtained with the highest LES resolution are very close to the DNS reference results (Figs. 7a,b), almost independent of the LES method used. The best results, both in terms of wave amplitude and total energy dissipation, were obtained with ALDM in the standard formulation. Since the spurious oscillations observed in some 3D LESs with ALDM did not occur in any of the 2.5D simulations, we do not present any results using the modified ALDM with 
With a grid coarsened in the 
The grid further coarsened in the ζ direction causes the CDS4 simulations to quickly break down. The ALDM and DSM simulations are stable, but the quality of the result is poor, showing too much total energy dissipation and wave amplitude decay.
d. Summary of case 1
The unstable IGW is the most complex test case presented here. It involves multiple breaking events, and the total time of the breaking is similar to the wave period. It is thus a challenging test for the LES methods in 3D and 2.5D. In 3D LESs, we obtained good agreement with the reference DNS using the DSM and ALDM (with 
The 3D and 2.5D LES results depend strongly on the numerical resolution in the ζ direction (of the wave phase), while the resolution in the 
6. Case 2: Stable inertia–gravity wave
a. Three-dimensional DNS
The reference DNS results are taken from Fruman et al. (2014). They presented simulations with 720 × 96 × 1024 cells and with 512 × 64 × 768 cells. To have at least a small ensemble of four members for comparison, we repeated these simulations (adding low-level white noise to the velocity components of the initial condition) running until t = 1 h. The ensemble average and standard deviation of these four simulations is shown in Fig. 8.
Statically stable IGW (3D). Base wave amplitude a and total dissipation rate 
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
The breaking of the wave is weaker than in the unstable IGW case, and it lasts only for a short period in time. The initial perturbations grow during the first minutes and generate some turbulence, which remains confined to the least stable part of the domain and is dissipated quickly. The dissipation peak occurs at t = 11 min, and, 30 min later, the turbulence has vanished completely.
b. Three-dimensional LES
The computational domain for the stable inertia–gravity wave is smaller in the 
We performed LESs using ALDM (
With the grid coarsened in the 
If the grid is not coarsened in the 
c. 2.5D simulations
The results of the 2.5D simulations (DNS and LES) are summarized in Fig. 9. As for the 3D LESs, we chose LES grids with high resolution (64 × 80 cells) and grids coarsened in the 
Statically stable IGW (2.5D). Base wave amplitude a and total dissipation rate 
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
The matching of the simulation results is very similar to the 3D cases. With the highest resolution (64 × 80 cells), the agreement is almost perfect, independent of the LES method used. With the grid coarsened in the 
d. Summary of case 2
The breaking of the stable IGW is weak and lasts only for a fraction of the wave period. Both in 3D and in 2.5D LESs, we obtained good agreement with the reference DNSs as long as we chose a comparatively high resolution in the ζ direction, while the results were not much affected by choosing a low resolution in the 
7. Case 3: Unstable high-frequency gravity wave
a. Three-dimensional DNS
Fruman et al. (2014) simulated the case of a breaking unstable HGW on grids with 15363 cells, 7683 cells, and 3843 cells. They found no notable differences between the two highest resolutions. We added another two simulations with 7683 cells and 3843 cells and averaged the results of these five DNSs. The results are presented in Fig. 10.
Statically unstable HGW (3D). Base wave amplitude a and total dissipation rate 
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
The wave breaking is much more intense than in both IGW cases. The generation of turbulence starts immediately after the initialization in the unstable part of the wave and is quickly advected also into the stable part because of the high phase velocity of the wave. At the time of maximum energy dissipation (around t = 15 min) turbulence is distributed almost homogeneously in the whole domain. The nondimensional wave amplitude rapidly decreases from an initial value of a = 1.2 to a ≈ 0.3 after 30 min and does not change significantly any more after that time. The breaking process is analyzed in more detail by Fruman et al. (2014).
b. Three-dimensional LES
The domain for the unstable high-frequency gravity wave case is almost cubic. In a number of LESs with different resolutions in the horizontal and the vertical directions, we could not find any indication that different resolutions in the different directions make a great deal of difference. Hence, we present here the results of three LES grids with coarse (203), medium (403), and fine (803) resolution (with the fine resolution corresponding to a cell size of 36.6 m × 37.5 m × 37.5 m). On the medium and fine grid, the initial perturbation is resolved almost perfectly (see Fig. 2c), while, on the coarse grid, there are some slight deviations in the initial perturbation energy distribution. We performed LESs on these grids using ALDM (
With the high LES resolution of 803 cells, the results are very similar to the DNS (Figs. 10a,b). The base wave amplitude decay is slightly overpredicted with ALDM and CDS4, but the amplitude remains almost within the variations among the DNS ensemble members. The peak dissipation rate matches well with the DNS in all cases. With CDS4, the dissipation falls off a bit too rapidly after the peak. With modified ALDM (
When the resolution is reduced to 403 cells (Figs. 10c,d), the main difference is in the CDS4 simulations. The turbulence during the peak of breaking is too strong, and the molecular dissipation is not sufficient on the coarse grid to keep the energy balance. Energy piles up at the smallest resolved wavenumbers (see the energy spectra in Fig. 12), and numerical errors lead to an increase of flow total energy, which eventually also affects the largest resolved scales and therefore the amplitude of the base wave. The time of simulation breakdown is almost the same in all ensemble members. By using the turbulence parameterization schemes, this instability can be avoided. The best matching with the DNS results is obtained with the original ALDM. Only about 10% of the peak energy dissipation is resolved (see Fig. 11a), but the sum of resolved molecular and numerical dissipation matches quite well with the DNS result. Also, the ratio between 
Statically unstable HGW (3D). Comparison of (a),(c),(e) resolved (
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
In Fig. 12, we present the energy spectra of all LESs with 403 cells compared to the DNS spectra. The CDS4 spectra are wrong, as mentioned above, and the method fails for this case. The ALDM and DSM spectra are very close to the DNS reference for wavelengths 
Statically unstable HGW (3D). Distributions of total, kinetic, and potential energy over vertical wavelength 
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
The results obtained with the coarsest grid, with 203 cells (Figs. 10e,f), are similar to those with the medium resolution. The simulations with CDS4 break down as a result of the unbounded growth of numerical errors. ALDM with 
c. 2.5D simulations
The results of the 2.5D simulations (DNS and LES) are summarized in Fig. 13. LES grids with high (802 cells), medium (402 cells), and coarse resolution (202 cells) were used. The same LESs were performed eight times with some low-level white noise superposed on the initial condition (consisting of the base wave and its leading primary perturbation) and once with no added noise. The results of these nine realizations were then averaged. The reference 2.5D DNSs were run at a resolution of 500 × 500 cells. An ensemble of six DNSs was used for the calculation of mean values and standard deviations.
Statically unstable HGW (2.5D). Base wave amplitude a and total dissipation rate 
Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0321.1
With the highest resolution (802 cells), the results in terms of wave amplitude and total dissipation rate are in very close agreement with the reference DNSs. Only for CDS4 is the dissipation rate a bit too low during the period of decreasing dissipation.
At the medium resolution (402 cells), the DSM and ALDM results are very similar and still in good agreement with the DNSs. The dissipation peak is slightly shifted to earlier times according to the dissipation acting at larger wavenumbers and the hence reduced time required for flow energy to reach this range. CDS4, however, predicts the wrong evolution of the wave amplitude and dissipation rate and cannot be recommended for this resolution.
At the coarsest resolution (202 cells), ALDM and DSM still do a very good job in predicting the amplitude decay and the dissipation maximum. The dissipation peak is further shifted forward in time because of the reduced time the flow energy needs to move through the spectrum. In the CDS4 simulation, however, the dissipation rate becomes negative after approximately 20 min, and, hence, the predicted flow field is completely wrong, although the simulations remain stable in a numerical sense during the whole simulated period.
d. Summary of case 3
The unstable HGW involves much stronger turbulence than the IGW cases, and thus the buoyancy forces are weaker compared to the acceleration associated with turbulent motions. The original ALDM and the DSM thus do an excellent job in predicting the dissipation rates and the wave amplitude decay over time, even at a very coarse resolution with a cell size of about 
8. Conclusions
We scrutinized different methods of large-eddy simulation for three cases of breaking monochromatic gravity waves. The methods tested included the following: the adaptive local deconvolution method (ALDM), an implicit turbulence parameterization; the dynamic Smagorinsky method (DSM); and a plain fourth-order central discretization without any turbulence parameterization (CDS4). The test cases have been carefully designed and set up by Remmler et al. (2013) and Fruman et al. (2014) based on the primary and secondary instability modes of the base waves and included an unstable and a stable inertia–gravity wave, as well as an unstable high-frequency gravity wave. All simulations presented were run in 2.5D and 3D domains, and, for all simulations, a small ensemble of simulations starting from slightly different initial conditions was performed in order to assess the sensitivity and robustness of the results.
The original ALDM leads to spurious oscillations of the buoyancy field in some 3D simulations, where the velocity field is very smooth for a long time. We thus developed a modified version of the ALDM flux function. The modification led to a significant reduction of the oscillations but also increased the overall energy dissipation.
For all three test cases, we started at an LES resolution of 80 cells per wavelength of the original wave and gradually reduced the resolution in all three directions. The inertia–gravity wave cases, in which the wave vector almost coincides with the vertical direction, were very sensitive to the resolution in the direction of the wave vector, while the resolution in the other directions could be strongly reduced without a massive negative effect on the overall results.
We found that results obtained with ALDM and DSM are generally in good agreement with the reference direct numerical simulations as long as the resolution in the direction of the wave vector is sufficiently high. The CDS4 simulations, without turbulence parameterization, are only successful if the resolution is high and the level of turbulence comparatively low. In cases with low turbulence intensity and a smooth velocity field for long time periods (unstable and stable IGW) ALDM generated spurious oscillations in the buoyancy field, which we could avoid by using the modified numerical flux function. However, this was not necessary in the case with a high turbulence level (unstable HGW) and in all 2.5D simulations.
Our results back the findings of Remmler and Hickel (2012, 2013, 2014), who showed that both DSM and ALDM are suitable tools for the simulation of homogeneous stratified turbulence. Applying the same methods to gravity wave breaking, where turbulence is spatially inhomogeneous and intermittent in time, reveals that DSM is, in some cases, more robust than ALDM, although ALDM provides a better approximation of the spectral eddy viscosity and diffusivity in homogeneous stratified turbulence (Remmler and Hickel 2014).
In all simulations, we observed that the peak of dissipation occurs earlier in simulations with coarser computational grids. This is more pronounced in 2.5D LESs but also apparent in 3D LESs. We explain this time difference by the time required for flow energy to move from the smallest resolved wavenumbers in an LES to the dissipative scales in a DNS. Among the tested LES methods, there is no method that can account for this time lag. However, the large-scale flow and the maximum dissipation can still be predicted correctly.
Fruman et al. (2014) have shown that, in some cases, 2.5D simulations can be sufficient to get a good estimate of the energy dissipation during a breaking event. We showed that, with ALDM and DSM, reliable results can be obtained in 2.5D simulations with fewer than 2000 computational cells. Such inexpensive simulations will allow for the running of large numbers of simulations in order to study the influence of various parameters on wave breaking, such as stratification, wavelength, amplitude, propagation angle, and viscosity. A possible automated approach would involve computing the growth rates of perturbations of the original waves, setting up an ensemble of 2.5D LESs initialized by the base wave and its leading primary perturbation, and extracting key data from the LES results, such as the maximum energy dissipation, the amplitude decay, and the duration of the breaking event. Another potential application of our findings is the (2.5D or 3D) simulation of wave packets in the atmosphere, which is computationally feasible only if small-scale turbulence remains unresolved and is treated by a reliable subgrid-scale parameterization, such as ALDM or DSM.
U. A. and S. H. thank Deutsche Forschungsgemeinschaft for partial support through the MetStröm (Multiple Scales in Fluid Mechanics and Meteorology) Priority Research Program (SPP 1276), and through Grants HI 1273/1-2 and AC 71/4-2. Computational resources were provided by the HLRS Stuttgart under Grant DINSGRAW.
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