## 1. Introduction

The parameterization of gravity waves (GWs) is of significant importance in atmospheric global circulation models (GCM), in global numerical weather prediction (NWP) models, and in ocean models. In spite of the increasing available computational power and the corresponding increase of spatial resolution of GCMs and NWP models, for the time being, an important range of GW spatial scales remains unresolved both in climate simulations and in global NWP (Alexander et al. 2010). Numerous studies indicate that a representation of GWs is necessary for a realistic description of various aspects of the middle-atmospheric circulation [e.g., the Brewer–Dobson circulation (Butchart 2014)] and hence the zonal-mean winds and temperature (Lindzen 1981; Houghton 1978), the quasi-biennial oscillation (QBO) (Holton and Lindzen 1972; Dunkerton 1997), sudden stratospheric warmings (SSW) (Richter et al. 2010; Limpasuvan et al. 2012), and—via feedback loops—the tropospheric circulation [e.g., the North Atlantic Oscillation (Scaife et al. 2005, 2012)].

Parameterizations of the gravity wave drag are indeed applied in most GCMs or NWP models (Lindzen 1981; Medvedev and Klaassen 1995; Hines 1997a,b; Lott and Miller 1997; Alexander and Dunkerton 1999; Warner and McIntyre 2001; Lott and Guez 2013). In some way or other they all use the Wentzel–Kramer–Brillouin (WKB) theory, but with some important simplifications: that is, (i) the assumption of a steady-state wave field and background flow, (ii) the neglect of the impact of horizontal large-scale flow gradients on the GWs, and (iii) one-dimensional vertical propagation. Under these conditions, the wave-dissipation or nonacceleration theorem states that GWs can deposit their momentum only where they break. In theoretical analyses of this problem in a rotating atmosphere, Bühler and McIntyre (1999, 2003, 2005) point out that the steady-state assumption can lead to the neglect of important aspects of the interaction between GWs and mean flow. By wave-resolving numerical simulations and analyses on the basis of a nonlinear Schrödinger equation, Dosser and Sutherland (2011) have demonstrated the relevance of direct GW–mean flow interactions as well. Still missing, however, is an explicit assessment of the significance of the direct interaction between transient GWs and mean flow as represented by WKB, called direct GW–mean flow interactions in the following. WKB modeling for diagnostic purposes, as by the Gravity-Wave Regional or Global Ray Tracer (GROGRAT) model (Marks and Eckermann 1995; Eckermann and Marks 1997), is a well-established tool (e.g., Eckermann and Preusse 1999), but such analyses leave out the GW impact on the large-scale flow. A semi-interactive approach to studies of the interaction between GWs and solar tides has been described by Ribstein et al. (2015), however, with a simplified treatment of the GW impact on the solar tides, using effective Rayleigh-friction and thermal-relaxation coefficients. The numerical implementation of a fully interactive WKB theory, allowing direct GW–mean flow interactions, is not a trivial task and should best be accompanied by validations against wave-resolving data. In a Boussinesq framework, the representation of direct GW–mean flow interactions by a WKB algorithm has been studied by Muraschko et al. (2015) for vertically propagating idealized wave packets with variable vertical wavenumber. WKB theory had been implemented there in a two-dimensional phase space spanned by the physical height and the vertical wavenumber. The phase-space representation (Bühler and McIntyre 1999; Hertzog et al. 2002; Broutman et al. 2004) turned out to be effective to avoid numerical instabilities due to caustics (i.e., when ray volumes representing GWs become collocated in physical space but have different vertical wavenumbers and thus group velocities). Muraschko et al. (2015) could demonstrate the validity of their approach by comparisons against wave-resolving large-eddy simulation (LES) data.

The Boussinesq setting, however, leaves out the amplitude growth experienced by atmospheric GWs due to propagation into altitudes with decreasing density. That process, however, is central for the ensuing wave breaking due to static or dynamic instability at large GW amplitudes. By the wave-dissipation theorem, steady-state GW parameterizations depend on wave breaking as the mechanism leading to a large-scale GW drag. How this mechanism—the only one represented by present GW parameterizations—competes with GW drag by direct GW–mean flow interactions and how well the latter can be captured in the atmosphere by a WKB algorithm have remained mainly unanswered questions to date. These are addressed here by investigations in a non-Boussinesq atmosphere, where the WKB algorithm is supplemented by a wave-breaking scheme. Steady-state WKB simulations are considered as well, representing the GW parameterization approach in present weather and climate models. Like Muraschko et al. (2015), we consider idealized cases of upwardly propagating horizontally homogeneous GW packets. LES provide wave-resolving reference data.

Our investigations are described as follows: the theoretical background of the work is presented in section 2, while the corresponding numerical models are introduced in section 3. This is followed in section 4 by the presentation of the results. In section 5, the main findings of the work are summarized, and conclusions are drawn.

## 2. Theoretical background

*x*–

*z*plane:

*g*is the gravitational constant,

*R*is the ideal gas constant,

*p*pressure and

*u*and

*w*are the velocity components in the

*x*–

*z*plane. We assume that the flow consists of a reference part constant in time as well as a large-scale background part and a small-scale wave part both changing in time.

### a. WKB theory

*x*direction. By linearization of Eqs. (1)–(4) about the reference and large-scale background, introducing the wave pressure

*N*denotes the Brunt–Väisälä frequency. In the polarization relation [Eq. (5)],

*U*

_{w},

*W*

_{w},

*B*

_{w}, and

*P*

_{w}denote WKB wave amplitudes of

*u*

_{w},

*w*

_{w},

*b*

_{w}, and

*p*

_{w}, where

*k*is a constant horizontal wavenumber, always assumed to be positive, and the local phase

*ϕ*defines the local vertical wavenumber

*z*and

*t*.

*field equations*[Eqs. (8) and (10)] is the ray technique, observing that along characteristic, so-called ray trajectories, defined by

*ray equations*:

*m*,

*β*, each having wavenumber and wave-action density

*m*and

*z*, one introduces a wave-action density

*δ*denoting the Dirac delta function. It can then be shown that

### b. Wave breaking

As WKB theory does not account for the possible turbulent wave breakdown at large GW amplitudes, the coupled ray and mean flow equations above have been supplemented with a saturation criterion in an attempt to add additional parameterization of this process. Comparisons between simulations with or without this “turbulence scheme” also enable an assessment of the relevance of wave breaking for the GW drag, as compared to the direct GW–mean flow interactions described by WKB.

*z*during the wave propagation (Lindzen 1981) so that somewhere within a complete wave cycle

*α*represents well-known uncertainties of the criterion Eq. (25). Stability analyses (e.g., Lombard and Riley 1996; Achatz 2005) and direct numerical simulations (Fritts et al. 2003, 2006; Achatz 2007; Fritts et al. 2009) indicate that GWs are unstable already below the static-instability threshold, and strongly nonhydrostatic, modulationally unstable wave packets also tend to break earlier (Dosser and Sutherland 2011). Another issue is that the criterion does not account for the possibility of destructive interference of different spectral components that would retard the onset of static instability.

*z*, turbulence is assumed to be generated that acts to damp wave-action density

In summary, the weakly nonlinear coupled GW–mean flow equations [Eqs. (11), (12), (20), and (24)] describe the time evolution of a *transient* GW field through a *transient* large-scale background flow in a *direct* manner. In addition, wave breaking is accounted for in the WKB models by applying the saturation criterion [Eq. (27)] and reducing the wave-action density proportionally to

### c. Steady-state WKB theory

*α*uncertainty parameter as explained in section 2b. Notably, this approach leads to instantaneous pseudomomentum-flux profiles. Variations of the boundary conditions at the source altitude are communicated immediately throughout the whole altitude range of a model, while in a more realistic transient approach any signal propagates at the group velocity. There are various possibilities of implementations of steady-state parameterizations (Fritts and Alexander 2003; Alexander et al. 2010): for example, by allowing a spectrum via a superposition of components as just described, each with its own values of vertical wavenumber and wave-action density at the source altitude. Lott and Guez (2013), for example, suggest picking these in a stochastic manner from a random sample. However, all of these approaches are instantaneous, and they only allow GW–mean flow interactions where GWs break.

## 3. Test cases and numerical models

*σ*defines the vertical size of the GW packet

*x*direction, the wave packet is initialized with infinite extent and a constant wavenumber

*k*. To initialize the idealized wave packet in the wave-resolving LES, the following perturbations are prescribed at initial time

Seven idealized test cases have been investigated. Three cases elaborate the refraction and the reflection of hydrostatic GW packets from a background jet, while four other cases aim to study static and modulational instability of hydrostatic and nonhydrostatic wave packets, including the process of wave breaking. The initial wave packet characteristics *dz* ≈ λ_{z0}/30, while the WKB simulations have been done at a vertical resolution of *dz* ≈ λ_{z0}/10: that is, at a resolution 3 times coarser than the reference LES (see further details in Tables 1 and 2). Both LES and WKB simulations with an increased resolution have been performed without observing significant changes in the results, which confirms that a convergence in the numerical results has been reached with the resolution described above.

Settings used in the idealized cases.

Settings used in the idealized cases.

### a. Reference LES model

The reference LES model called PincFloit solves the pseudo-incompressible equations [i.e., a soundproof approximation of the Euler equations [Eqs. (1)–(4)] (Durran 1989)]. A third-order Runge–Kutta time scheme and a finite-volume spatial discretization are applied, which involve an adaptive local deconvolution model (ALDM) (Hickel et al. 2006) for turbulence parameterization. Alternatively, the Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL; van Leer 1979) can also be used in the finite-volume scheme. Tests using both schemes for our cases did not show an important sensitivity. It is important to mention that, in contrast to the WKB simulations, the reference LES is fully nonlinear and enables the description of wave–wave interactions as well as turbulent wave dissipation, which, with a high resolution, implies a realistic description of compressible flows. The PincFloit model has been described and tested in detail by Rieper et al. (2013b).

### b. Eulerian WKB model

The Eulerian implementation of the WKB equations solves the flux form [Eq. (17)] of the phase-space wave-action density equation using the MUSCL finite-volume discretization on the *z*–*m* plane, with an equidistant staggered grid in both the *z* and *m* direction. As a consequence, the phase-space wave-action density

### c. Lagrangian WKB model

*m*is given by

*m*direction. Because of the conserved area

*a*, the evolution

*i*of the Lagrangian WKB model, with

*nz*being the number of vertical levels, and

*j*th ray volume relevant to layer

*i*, and

*i*. The term

*j*th ray volume in the

*m*direction so that

*j*th ray being in the

*i*th vertical layer, as ray volumes might overlap with several vertical layers (see Fig. 1). Using the dispersion relation [Eq. (6)], one obtains the following from Eq. (49):

*i*th layer, the wave amplitude is reduced following Eq. (30), with

### d. Steady-state WKB model

A numerical model based on the steady-state WKB theory (section 2c) has been implemented as well, in order to enable a comparison with the transient WKB simulations and thus an assessment of present-day GW parameterizations. For optimal correspondence between transient and steady-state simulations, the time-dependent boundary values for wavenumber and wave-action density at the “source” altitude

## 4. Results

We first give a comparative validation of the Eulerian and the Lagrangian WKB models, demonstrating the superiority of the latter. We then discuss a case where no wave breaking is active but where the negligence of direct GW–mean flow interactions would make a fundamental difference. Finally, we compare the relative importance of direct GW–mean flow interactions and of wave breaking in cases where both are active. There we also demonstrate the limitations of a steady-state approach with wave-breaking scheme.

### a. Comparative validation of the Eulerian and Lagrangian WKB models

The refraction of a hydrostatic GW packet (

*z*and

*m*directions, a good agreement with the LES can be achieved (not shown); however, at the same time the computational time gets exhaustive (i.e., 2–3 times larger than the computing time of the LES). It is to be mentioned here that, even by using the same vertical resolution in the Eulerian and the Lagrangian transient WKB simulations, the latter is more efficient computationally than the former, by a factor of 10–100, depending on the number of rays used in the Lagrangian model. The better efficiency of the Lagrangian model is due to the fact that, in this framework, (i) there is no necessity to span the whole phase-space volume, including all its regions where the wave-action density is negligibly small, and (ii) the prediction of the latter is done by solving the trivial conservation equation [Eq. (20)]. In contrast, in the Eulerian model, the prognostic equation [Eq. (17)] is solved using the MUSCL finite-volume scheme, which requires a relatively expensive reconstruction of the fluxes on the cell edges. These findings regarding the efficiency of the transient WKB simulations motivated the use of the Lagrangian model in all our further studies.

### b. Role of direct GW–mean flow coupling

Initializing a hydrostatic GW packet with somewhat shorter horizontal wavelength (

### c. Role of wave breaking

Finally, a set of cases with unstable GW packets or with GW packets turning into unstable regimes has also been studied. Comparisons between LES and transient WKB simulations with wave-breaking parameterizations serve to validate the latter. More importantly, however, these cases are to provide an assessment of the comparative importance of direct GW–mean flow interactions, as represented by the WKB model without wave-breaking parameterization, and the wave-breaking process itself, as represented by that latter turbulence scheme.

In the first case, a Gaussian-shaped hydrostatic wave packet (

The evolution of a Gaussian-shaped nonhydrostatic GW packet (*α* compared to the previous case might be due to the added inclination of nonhydrostatic wave packets to become modulationally unstable. By looking at the vertical structures of the wave energy and the induced mean wind in Figs. 6a–l, one finds again that the wave-breaking parameterization provides only small corrections on top of the relatively good results provided by the transient WKB simulations without the saturation scheme.

The next case involves a cosine-shaped nonhydrostatic GW packet (

Finally, a hydrostatic GW packet (*α* suggests that, in case of a critical layer, GWs tend to break as predicted by classic static-instability criteria.

Our results suggest that wave breaking is of secondary importance in comparison with the direct GW–mean flow interactions even for large-amplitude GWs. Since present-day GW-drag parameterizations exclusively rely on wave breaking, the question arises what results they would yield in the cases considered. Therefore, the cases STIH, STINH, and MI have been also been simulated using the steady-state WKB model based on sections 2c and 3d. Figure 9 shows the corresponding results for case STINH, which are to be compared with Fig. 6, where the transient WKB and the LES results are shown. The integrated energy shows that the wave energy is overdamped and that the kinetic energy of the mean flow is strongly underestimated in the steady-state WKB model. The former is also observed in the Hovmöller diagram of the wave energy. The Hovmöller diagram of the induced mean wind shows that the magnitude of the GW drag is too small in the steady-state WKB model, and also its structure is very different from that of the LES and the transient WKB simulation. One should, of course, restrict the comparison between the models to the vertical region above the source (

## 5. Summary and conclusions

The steady-state approximation to WKB theory used nowadays in GW-drag parameterizations implies that the only GW forcing on the mean flow is due to wave breaking. Transient GW–mean flow interactions can, however, act as another important coupling mechanism. This study provides an assessment of the comparative importance of these processes in typical atmospheric situations, albeit idealized. Focusing on single-column scenarios for the time being, considered GW packets are horizontally homogeneous, and the mean flow has only a vertical spatial dependence. The wave scales and amplitudes, however, are representative, although not of inertia–gravity waves affected by rotation. Fully interactive transient WKB simulations are used to describe the simultaneous development of GWs and mean flow. All of these simulations are validated against wave-resolving LES, thereby assessing the reliability of the methods employed.

The WKB algorithms used allow the simulation of transient GW development. In both variants, Eulerian or Lagrangian, the mean flow is fully coupled to the wave field. This is enabled by a spectral approach, employing wave-action density in position-wavenumber phase space, the key to avoiding otherwise detrimental numerical instabilities due to caustics. The Eulerian approach spans the whole phase space. It thus quickly tends to be expensive, often more so than LES. The Lagrangian ray-tracing approach, however, focuses on regions of phase space with nonnegligible wave action. This makes it considerably more efficient, by orders of magnitude, than the wave-resolving simulations. Certainly this might change in situations where broad spectra develop. So far, however, we have not met with such a case.

A systematic investigation of the comparative relevance of wave breaking, as compared to direct GW–mean flow interactions, has been enabled by the implementation of a simple turbulence scheme. Turbulence is invoked whenever the wave field has the possibility to become statically unstable. A flux-gradient parameterization of turbulent fluxes is used, by way of eddy viscosity and diffusivity. The ensuing damping of the GW field is hence scale selective so that small scales are damped more strongly. Generally, it is found, by comparison against the LES data, that the static-stability criterion tends to generate turbulence too quickly. This might be explained by phase cancellations between different spectral components so that higher amplitudes are required to really lead to the onset of turbulence. Nonetheless, the turbulence scheme works quite well if validated against the LES simulations.

Finally also a steady-state WKB model has been implemented, representing the approach in current GW-drag parameterizations. Caustics are not an issue in such a context so that a spectral formulation is not necessary. The simulations discussed here consider locally monochromatic GW fields that are damped to the saturation limit once static instability is diagnosed. Spectral extensions have been considered as well (not shown), but these did not yield substantially different results.

The pattern observed in our simulations is quite clear: whenever a wave impact on the mean flow is observed, the direct GW–mean flow interactions dominate over the wave-breaking effect. It is important that these interactions depend on wave transience. Without the latter they would not be possible—because of the nonacceleration paradigm—without onset of turbulence. Interesting effects arise that would be missed by a steady-state scheme. An example is partial reflection from a jet that would not occur within linear theory. The wave-induced wind contributes sufficiently so that part of the GW packet substantially changes its propagation. In the various turbulent cases considered, be it apparent wave breaking by direct static instability or triggered by modulational instability, we see a dominant impact from direct GW–mean flow interactions as well. Even when the turbulence scheme is suppressed, the results between transient WKB and LES agree to leading order, at least as far as the spatial distribution of wave energy and mean wind are concerned. Turbulence acts to next order and ensures the correct dissipation of total energy.

Turbulence without direct GW–mean flow interactions, however, fails to explain the LES data: the steady-state WKB simulations exhibit way too strong damping of the GW energy, and also the mean flow is underestimated significantly. This argues for a serious attempt at including GW-transience effects into GW-drag parameterizations. Undoubtedly, the implementation of transient WKB into climate and weather models will face considerable efficiency issues. The strong role we see for direct GW–mean flow interactions could provide motivation, however, to overcome these.

A final caveat might be that, notwithstanding the apparent success of transient WKB seen here, there are plenty of cases where this approach might also find its limitations. Not only have we neglected lateral GW propagation, an effect known to also be potentially important (Bühler and McIntyre 2003; Senf and Achatz 2011; Ribstein et al. 2015), but the limits of WKB as a whole will be reached where strong wave–wave interactions come into play, or where significant small-scale structures are present (Fritts et al. 2013).

## Acknowledgments

U.A. and B.R. thank the German Federal Ministry of Education and Research (BMBF) for partial support through the program Role of the Middle Atmosphere in Climate (ROMIC) and through Grant 01LG1220A. U.A. and J.W. thank the German Research Foundation (DFG) for partial support through the research unit Multiscale Dynamics of Gravity Waves (MS-GWaves) and through Grants AC 71/8-1, AC 71/9-1, and AC 71/10-1.

## REFERENCES

Achatz, U., 2005: On the role of optimal perturbations in the instability of monochromatic gravity waves.

,*Phys. Fluids***17**, 094107, doi:10.1063/1.2046709.Achatz, U., 2007: Gravity-wave breaking: Linear and primary nonlinear dynamics.

,*Adv. Space Res.***40**, 719–733, doi:10.1016/j.asr.2007.03.078.Achatz, U., R. Klein, and F. Senf, 2010: Gravity waves, scale asymptotics and the pseudo-incompressible equations.

,*J. Fluid Mech.***663**, 120–147, doi:10.1017/S0022112010003411.Alexander, M. J., and T. J. Dunkerton, 1999: A spectral parameterization of mean-flow forcing due to breaking gravity waves.

,*J. Atmos. Sci.***56**, 4167–4182, doi:10.1175/1520-0469(1999)056<4167:ASPOMF>2.0.CO;2.Alexander, M. J., and Coauthors, 2010: Recent developments in gravity-wave effects in climate models and global distribution of gravity-wave momentum flux from observations and models.

,*Quart. J. Roy. Meteor. Soc.***136**, 1103–1124, doi:10.1002/qj.637.Becker, E., 2004: Direct heating rates associated with gravity wave saturation.

,*J. Atmos. Sol.-Terr. Phys.***66**, 683–696, doi:10.1016/j.jastp.2004.01.019.Bretherton, F. P., 1966: The propagation of groups of internal gravity waves in a shear flow.

,*Quart. J. Roy. Meteor. Soc.***92**, 466–480, doi:10.1002/qj.49709239403.Broutman, D., J. W. Rottman, and S. D. Eckermann, 2004: Ray methods for internal waves in the atmosphere and ocean.

,*Annu. Rev. Fluid Mech.***36**, 233–253, doi:10.1146/annurev.fluid.36.050802.122022.Bühler, O., and M. E. McIntyre, 1999: On shear-generated gravity waves that reach the mesosphere. Part II: Wave propagation.

,*J. Atmos. Sci.***56**, 3764–3773, doi:10.1175/1520-0469(1999)056<3764:OSGGWT>2.0.CO;2.Bühler, O., and M. E. McIntyre, 2003: Remote recoil: A new wave-mean interaction effect.

,*J. Fluid Mech.***492**, 207–230, doi:10.1017/S0022112003005639.Bühler, O., and M. E. McIntyre, 2005: Wave capture and wave-vortex duality.

,*J. Fluid Mech.***534**, 67–95, doi:10.1017/S0022112005004374.Butchart, N., 2014: The Brewer–Dobson circulation.

,*Rev. Geophys.***52**, 157–184, doi:10.1002/2013RG000448.Coiffier, J., 2011:

2nd ed. Cambridge University Press, 340 pp.*Numerical Weather Prediction.*Dosser, H. V., and B. Sutherland, 2011: Anelastic internal wave packet evolution and stability.

,*J. Atmos. Sci.***68**, 2844–2859, doi:10.1175/JAS-D-11-097.1.Dunkerton, T. J., 1997: The role of gravity waves in the quasi-biennial oscillation.

,*J. Geophys. Res.***102**, 26 053–26 076, doi:10.1029/96JD02999.Durran, D. R., 1989: Improving the anelastic approximation.

,*J. Geophys. Res.***46**, 1453–1461, doi:10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2.Eckermann, S. D., and C. Marks, 1997: GROGRAT: A new model of the global propagation and dissipation of atmospheric gravity waves.

,*Adv. Space Res.***20**, 1253–1256, doi:10.1016/S0273-1177(97)00780-1.Eckermann, S. D., and P. Preusse, 1999: Global measurements of stratospheric mountain waves from space.

,*Science***286**, 1534–1537, doi:10.1126/science.286.5444.1534.Fritts, D. C., and M. J. Alexander, 2003: Gravity wave dynamics and effects in the middle atmosphere.

,*Rev. Geophys.***41**, 1003, doi:10.1029/2001RG000106.Fritts, D. C., C. Bizon, J. A. Werne, and C. K. Meyer, 2003: Layering accompanying turbulence generation due to shear instability and gravity-wave breaking.

,*J. Geophys. Res.***108**, 8452, doi:10.1029/2002JD002406.Fritts, D. C., S. L. Vadas, K. Wan, and J. A. Werne, 2006: Mean and variable forcing of the middle atmosphere by gravity waves.

,*J. Atmos. Sol.-Terr. Phys.***68**, 247–265, doi:10.1016/j.jastp.2005.04.010.Fritts, D. C., L. Wang, J. A. Werne, T. Lund, and K. Wan, 2009: Gravity wave instability dynamics at high Reynolds numbers. Part I: Wave field evolution at large amplitudes and high frequencies.

,*J. Atmos. Sci.***66**, 1126–1148, doi:10.1175/2008JAS2726.1.Fritts, D. C., L. Wang, and J. A. Werne, 2013: Gravity wave–fine structure interactions. Part I: Influences of fine structure form and orientation on flow evolution and instability.

,*J. Atmos. Sci.***70**, 3710–3734, doi:10.1175/JAS-D-13-055.1.Grimshaw, R., 1975: Nonlinear internal gravity waves in a rotating fluid.

,*J. Fluid Mech.***71**, 497–512, doi:10.1017/S0022112075002704.Hertzog, A., C. Souprayen, and A. Hauchecorne, 2002: Eikonal simulations for the formation and the maintenance of atmospheric gravity wave spectra.

,*J. Fluid Mech.***107**, 4145, doi:10.1029/2001JD000815.Hickel, S., N. A. Adams, and J. Domaradzki, 2006: An adaptive local deconvolution model for implicit LES.

,*J. Comput. Phys.***213**, 413–436, doi:10.1016/j.jcp.2005.08.017.Hines, C. O., 1997a: Doppler spread gravity-wave parametrization of momentum deposition in the middle atmosphere. Part 1: Basic formulation.

,*J. Atmos. Sol.-Terr. Phys.***59**, 371–386, doi:10.1016/S1364-6826(96)00079-X.Hines, C. O., 1997b: Doppler spread gravity-wave parametrization of momentum deposition in the middle atmosphere. Part 2: Broad spectra and quasi monochromatic spectra, and implementation.

,*J. Atmos. Sol.-Terr. Phys.***59**, 387–400, doi:10.1016/S1364-6826(96)00080-6.Holton, J. R., and R. S. Lindzen, 1972: An updated theory for the quasi-biennial cycle of the tropical stratosphere.

,*J. Atmos. Sci.***29**, 1076–1080, doi:10.1175/1520-0469(1972)029<1076:AUTFTQ>2.0.CO;2.Houghton, J. T., 1978: The stratosphere and mesosphere.

,*Quart. J. Roy. Meteor. Soc.***104**, 1–29, doi:10.1002/qj.49710443902.Kim, Y.-J., D. S. Eckermann, and H. Chund, 2003: An overview of past, present and future gravity-wave drag parametrization for numerical climate and weather prediction models.

,*Atmos.–Ocean***41**, 65–98, doi:10.3137/ao.410105.Limpasuvan, V., J. H. Richter, Y. Orsolinic, F. Stordald, and O. Kvisseld, 2012: The roles of planetary and gravity waves during a major stratospheric sudden warming as characterized in WACCM.

,*J. Atmos. Sol.-Terr. Phys.***78–79**, 84–98, doi:10.1016/j.jastp.2011.03.004.Lindzen, R. S., 1981: Turbulence and stress owing to gravity wave and tidal breakdown.

,*J. Geophys. Res.***86**, 9707–9714, doi:10.1029/JC086iC10p09707.Lombard, P., and J. Riley, 1996: Instability and breakdown of internal gravity waves. I. Linear stability analysis.

,*Phys. Fluids***8**, 3271–3287, doi:10.1063/1.869117.Lott, F., and M. Miller, 1997: A new subgrid-scale orographic drag parametrization: Its formulation and testing.

,*Quart. J. Roy. Meteor. Soc.***123**, 101–127, doi:10.1002/qj.49712353704.Lott, F., and L. Guez, 2013: A stochastic parameterization of the gravity waves due to convection and its impact on the equatorial stratosphere.

,*J. Geophys. Res. Atmos.***118**, 8897–8909, doi:10.1002/jgrd.50705.Marks, C., and S. Eckermann, 1995: A three-dimensional nonhydrostatic ray-tracing model for gravity waves: Formulation and preliminary results for the middle atmosphere.

,*J. Atmos. Sci.***52**, 1959–1984, doi:10.1175/1520-0469(1995)052<1959:ATDNRT>2.0.CO;2.Medvedev, A. S., and G. P. Klaassen, 1995: Vertical evolution of gravity wave spectra and the parametrization of associated gravity wave drag.

,*J. Geophys. Res.***100**, 25 841–25 854, doi:10.1029/95JD02533.Muraschko, J., M. Fruman, U. Achatz, S. Hickel, and Y. Toledo, 2015: On the application of Wentzel–Kramer–Brillouin theory for the simulation of the weakly nonlinear dynamics of gravity waves.

,*Quart. J. Roy. Meteor. Soc.***141**, 676–697, doi:10.1002/qj.2381.Nappo, J. C., 2002:

1st ed. Academic Press, 276 pp.*An Introduction to Atmospheric Gravity Waves.*Ribstein, B., U. Achatz, and F. Senf, 2015: The interaction between gravity waves and solar tides: Results from 4-D ray tracing coupled to a linear tidal model.

,*J. Geophys. Res. Space Phys.***120**, 6795–6817, doi:10.1002/2015JA021349.Richter, J. H., F. Sassi, and R. R. Garcia, 2010: Towards a physically based gravity wave source parameterization.

,*J. Atmos. Sci.***67**, 136–156, doi:10.1175/2009JAS3112.1.Rieper, F., U. Achatz, and R. Klein, 2013a: Range of validity of an extended WKB theory for atmospheric gravity waves: One-dimensional and two-dimensional case.

,*J. Fluid Mech.***729**, 330–363, doi:10.1017/jfm.2013.307.Rieper, F., S. Hickel, and U. Achatz, 2013b: A conservative integration of the pseudo-incompressible equations with implicit turbulence parameterization.

,*Mon. Wea. Rev.***141**, 861–886, doi:10.1175/MWR-D-12-00026.1.Scaife, A. A., J. R. Knight, G. K. Vallis, and C. K. Folland, 2005: A stratospheric influence on the winter NAO and North Atlantic surface climate.

,*Geophys. Res. Lett.***32**, L18715, doi:10.1029/2005GL023226.Scaife, A. A., and Coauthors, 2012: Climate change projections and stratosphere–troposphere interaction.

,*Climate Dyn.***38**, 2089–2097, doi:10.1007/s00382-011-1080-7.Senf, F., and U. Achatz, 2011: On the impact of middle-atmosphere thermal tides on the propagation and dissipation of gravity waves.

,*J. Geophys. Res.***116**, D24110, doi:10.1029/2011JD015794.Sutherland, B. R., 2006: Weakly nonlinear internal gravity wavepackets.

,*J. Fluid Mech.***569**, 249–258, doi:10.1017/S0022112006003016.Sutherland, B. R., 2010:

*Internal Gravity Waves*. Cambridge University Press, 377 pp.van Leer, B., 1979: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method.

,*J. Comput. Phys.***32**, 101–136, doi:10.1016/0021-9991(79)90145-1.Warner, C. D., and M. E. McIntyre, 2001: An ultrasimple spectral parametrization for nonorographic gravity waves.

,*J. Atmos. Sci.***58**, 1837–1857, doi:10.1175/1520-0469(2001)058<1837:AUSPFN>2.0.CO;2.