## 1. Introduction

As one of the most fundamental physical modes in meteorology, gravity waves (GWs) are ubiquitous buoyancy oscillations in the atmosphere. The sources of excited gravity waves include, among others, topographic forcing (Smith 1980; Menchaca and Durran 2017), convection (Alexander et al. 1995; Lane et al. 2001), the jets (Zhang 2004; Plougonven and Zhang 2014; Hien et al. 2018), frontal systems (Snyder et al. 1993; Griffiths and Reeder 1996), and shear instability (Bühler et al. 1999; Bühler and McIntyre 1999). GW dynamics is also influenced by rotation, especially when the respective waves have relatively long horizontal wavelengths and short vertical wavelengths. These are called inertia–gravity waves (IGWs), and they are characterized by parcel oscillations that are elliptical instead of straight lines in the pure gravity wave case without the Earth rotation effect (Gill 1982; Nappo 2002). In addition to that, IGWs have relatively low intrinsic frequencies and small vertical group velocities. Many studies have documented the signals and the life cycles of IGWs associated with complex background flows, using observations (Wang and Geller 2003; Plougonven et al. 2003; Gong et al. 2012), numerical investigations of observed cases (Zhang et al. 2013, 2015), and idealized simulations (Wei and Zhang 2014, 2015; Wei et al. 2016).

GWs play a significant role in atmospheric dynamics on various spatial and temporal scales. For example, GWs can generate and modulate atmospheric turbulence (Shapiro 1980; Lane et al. 2004), and they can also initiate and organize moist convection (Zhang et al. 2001; Lane and Zhang 2011). As an important candidate, GWs can also potentially contribute significantly to building the atmospheric energy spectra (Callies et al. 2014; Bierdel et al. 2016; Sun et al. 2017). GWs are found to likely link the small-scale small-amplitude initial error to the rapid upscale error growth and thus limit mesoscale predictability (Zhang et al. 2007; Sun and Zhang 2016; Bierdel et al. 2018). The most important impact globally is because GWs can travel over large distances from their sources and transfer significant amounts of momentum and energy to high altitudes, which contributes to the forcing of the circulation and the variability of the middle atmosphere (Holton and Lindzen 1972; Houghton 1978; Lindzen 1981; Dunkerton 1997; Richter et al. 2010; Limpasuvan et al. 2012; Butchart 2014). The dynamics of the middle atmosphere can influence the tropospheric circulation by downward control (Haynes et al. 1991), and it can be very important for the forecasting of weather (Baldwin and Dunkerton 2001) and climate (Scaife et al. 2005, 2012).

Despite the increasing computational power, an important range of GW spatial scales remains unresolved in most atmospheric global circulation models (GCM) or in global numerical weather prediction (NWP) models (Alexander et al. 2010), and GW parameterizations are still applied in those models (Medvedev and Klaassen 1995; Hines 1997a,b; Lott and Miller 1997; Alexander and Dunkerton 1999; Warner and McIntyre 2001; Lott and Guez 2013). Most of the parameterization schemes are based on the Wentzel–Kramer–Brillouin (WKB) theory, but with some oversimplifications, including the assumption of a steady-state wave field and background flow, instantaneous GW propagation, and one-dimensional vertical propagation. Those abovementioned oversimplifications are often made to ensure numerical stability and model efficiency, but they can lead to the neglect of important aspects of the interaction between GWs and mean flow (Bühler and McIntyre 1999, 2003, 2005; Dosser and Sutherland 2011; Bölöni et al. 2016). The spectral phase-space representation (Bühler and McIntyre 1999; Hertzog et al. 2002) turns out to be effective to avoid numerical instabilities due to caustics. The issue of caustics occurs when by wave–mean flow interactions a wave field loses its initial locally monochromatic character (Tabaei and Akylas 2007; Rieper et al. 2013a). Based on the spectral approach, a prognostic Lagrangian WKB GW ray-tracing model has been implemented, using a multidimensional phase space spanned by physical space and the wavenumber space, and it has been successfully validated against wave-resolving large-eddy simulation (LES) data in different idealized settings, including one-dimensional (1D) vertically propagating idealized wave packets with variable vertical wavenumbers in a nonrotating Boussinesq atmosphere (Muraschko et al. 2015), 1D wave packets propagating in a nonrotating compressible atmosphere (Bölöni et al. 2016), and two-dimensional (2D) wave packets of parameterized submesoscale GWs interacting with resolved mesoscale GWs in a rotating Boussinesq atmosphere (Wilhelm et al. 2018). In addition, this approach has also been used to study the interaction between GWs and solar tides (Ribstein et al. 2015; Ribstein and Achatz 2016).

However, there is still no corresponding detailed validation of an IGW parameterization in a rotating compressible atmosphere against data from idealized wave-resolving simulations. This gives one of the motivations for the current study. More importantly, there are actually two available approaches for IGW parameterization, here called pseudomomentum scheme and direct scheme, respectively. The pseudomomentum scheme exploits the fact that in a Lagrangian-mean reference frame the effect of GWs on the large-scale flow only appears in the momentum equation (Andrews and McIntyre 1978). Application of this theory to an Eulerian-mean reference frame leads to a pseudomomentum-flux convergence by which the large-scale momentum is to be forced (Andrews and McIntyre 1976, 1978). As will be shown below this is at least justified if the large-scale flow is in geostrophic and hydrostatic balance. The direct scheme does not rely on any balance assumption with regard to the large-scale flow, and the large-scale flow is forced by anelastic momentum-flux convergence in the momentum equation, an elastic term also in the momentum equation, and entropy-flux convergence in the entropy equation, as given by Grimshaw (1975) and Achatz et al. (2017). All present-day operational IGW parameterizations represent, one way or other, simplified versions of the pseudomomentum approach, where the vertical gradient of pseudomomentum-flux convergence forces the resolved flow, when wave dissipation occurs (Fritts and Alexander 2003; Kim et al. 2003), and neither elastic nor thermal effects are taken into account. In contrast, to the best of our knowledge, there is still no numerical application of the direct scheme. This further motivates us to understand the differences between these two available schemes, to investigate which scheme performs better in validations against data from idealized wave-resolving simulations, and to verify how trustworthy the pseudomomentum approach is when the large-scale flow is unbalanced.

This article is arranged as follows. The theory part related to the two available approaches for the IGW parameterization will be shown in section 2, followed in section 3 by a brief introduction to the numerical models, and in section 4 by a presentation of the various cases used. Section 5 will present the budget analysis of the wave-induced forcing terms from the two different schemes in different idealized wave packet profiles. Based on the prognostic Lagrangian WKB GW ray-tracing model, both schemes will be verified against the data from wave-resolving simulations in section 6. In section 7, sensitivity experiments are performed to further investigate the relative importance of each wave-induced forcing term, as well as the wave–mean flow interactions during the wave propagation. Section 8 contains a summary.

## 2. Theory

For an explanation of the theoretical underpinnings of the two respective approaches we follow the presentation of Achatz et al. (2017) where, expanding on previous work by Grimshaw (1975), the theory is discussed mostly in nondimensional form. We translate the essentials into dimensional form and choose, for easier tractability, a heuristic formulation. For all mathematical details, the reader is referred back to Achatz et al. (2017).

*f*plane, (e.g., Durran 1989), with Coriolis parameter

*f*, without external sources or sinks,

**u**and

*w*are the horizontal and vertical components of the total wind

**v**, respectively;

*R*the ideal gas constant of dry air;

*θ*is potential temperature;

*π*is the Exner pressure; and

*g*is the gravitational acceleration.

*ϕ*, yielding local wavenumber

*ε*, and higher harmonics of the wave, induced by nonlinear interactions.

*f*and characteristic temperature

*k*,

*l*, and

*m*are the wavenumber components in

*x*,

*y*, and

*z*directions, respectively, and

*if*(i) in the entropy equation, (25), the entropy fluxes are neglected and (ii) the anelastic momentum-flux convergence and the elastic term in the horizontal-momentum equation, (26), are replaced by pseudomomentum-flux convergence, that is,

*provided the synoptic-scale flow is horizontal, geostrophic, and hydrostatic*. Hence the dynamics will obey (28) and the synoptic-scale flow will develop in a realistic manner. The corresponding calculations are nearly the same, only a bit simpler, as those leading from the original equations shown in (25) and (26)–(28), described by Achatz et al. (2017), and are therefore not detailed here any further. Therefore a recipe for an IGW parameterization can be to implement it exclusively into the horizontal-momentum equation, via pseudomomentum-flux convergence (Andrews and McIntyre 1976). In a single-column approximation, where all horizontal flux gradients are neglected, this is what present-day operational IGW parameterizations do (e.g., Alexander and Dunkerton 1999; Warner and McIntyre 2001; Scinocca 2002, 2003; Orr et al. 2010), and this pseudomomentum approach guarantees that a geostrophically and hydrostatically balanced flow is affected correctly by parameterized IGWs. However, the basis for this approach is even more fundamental. In their seminal work Andrews and McIntyre (1978) show that in a Lagrangian-mean reference frame the effect of arbitrary-amplitude GWs on a large-scale flow indeed only occurs in the momentum equation, by an appropriate forcing. Nonetheless an issue remains how to transfer this result to the Eulerian-mean reference frame that atmospheric models use, both for weather and climate applications. Andrews and McIntyre (1976), in the derivation of their generalized Eliassen–Palm-flux convergence for the zonal-mean zonal wind equation, use a scaling where weak wave amplitudes are assumed, and where hence approximate geostrophy and hydrostaticity of the mean flow allow the derivation of an approximate thermal-wind balance that can be used to avoid GW entropy-flux convergence and its application in a mean-flow entropy equation. Because of the weak-amplitude assumption advection of zonal momentum by the residual mean flow turns out to be negligible as well, so that one is left with a direct acceleration of the zonal-mean zonal wind by a the convergence of pseudomomentum (or generalized Eliassen–Palm) flux. The reliance of the pseudomomentum-flux result of Achatz et al. (2017) on large-scale balance appears to be directly related to this, however without the need for weak wave amplitudes. The following question may be asked: what happens if the required balance conditions are not fulfilled anymore? Although the theory of Achatz et al. (2017) exploits synoptic scaling, and hence derives geostrophic and hydrostatic balance, one can see (25) and (26) as intermediate results not explicitly relying on the large-scale flow being balanced, while the derivation of (28) makes heavy use of large-scale flow balance. It might therefore be more realistic to use in a more direct approach the unmodified entropy-flux convergence, anelastic momentum-flux convergence and elastic term. This could be the case, for example, if large-scale flow at different scales are of interest, or also ageostrophic flow such as the residual circulation. A central goal of the present work is a comparison of the two approaches in validations against wave-resolving simulations, and it will be shown that the direct approach is considerably more reliable.

*i*being

*x*or

*y*, and

*j*being

*x*,

*y*, or

*z*, can be related to its corresponding pseudomomentum-flux component

^{2}or 2 × 10

^{2}lead to a factor 2 or 5, respectively. Also of interest is that in some cases certain contributions to the pseudomomentum flux vanish, that is,

## 3. Description of the numerical models

In this section, the numerical code used for the validation cases is described, used either in a wave-resolving high-resolution mode with a subgrid-scale (SGS) turbulence parameterization to provide reference data or in a low-resolution mode with a WKB module switched on for validations of the two parameterization approaches.

### a. PincFloit LES

*f*plane the pseudoincompressible equations of Durran (1989) in flux form (Klein 2009; Rieper et al. 2013b):

*ρ*and

*η*is the dynamic shear viscosity coefficient,

*Q*is the total heating, here zero in all wave-resolving simulations. The elliptic problem for the pressure is solved using the hypre package (Falgout et al. 2006).

### b. PincFloit/MS-GWaM

**c**

_{g}and

**u**and stratification

*N*

^{2}, where from here on we denote the total resolved flow by lowercase letters. If coupled to MS-GWaM, PincFloit is used without molecular and turbulent viscosities, since IGWs should lead to the leading SGS effects.

*direct approach*the IGW impact on the resolved flow is mediated by anelastic momentum-flux convergence, the elastic term, and by entropy-flux convergence. Since in pseudoincompressible dynamics the nondiffusive continuity equation, that is, (41) with

*pseudomomentum approach*only the momentum equation is forced by the IGWs, via pseudomomentum-flux convergence leading to the replacements

At every Runge–Kutta step, information is exchanged between IGWs and large-scale flow dynamics. MS-GWaM determines the wave-induced fluxes based on either direct or pseudomomentum scheme, and updates the resolved large-scale flow in PincFloit. After integrating the PincFloit model, the new resolved large-scale flow information is given to MS-GWaM, which solves the ray-tracing equations and yields updated parameterized wave-induced fluxes, thus closing the circle. Within the last step we also use a saturation scheme for mimicking turbulent wave breakdown. Whenever the local wave amplitudes are large enough to make the wave field statically unstable, a turbulent viscosity is switched on that reduces the wave amplitudes within one time step to the saturation level. More details on the methods used by MS-GWaM can be found in Muraschko et al. (2015), Bölöni et al. (2016), and Wilhelm et al. (2018).

## 4. Wave packet validation cases

*T*= 240 K) atmosphere at rest is initially superposed to a locally monochromatic Gaussian IGW packet. Therein,

*y*direction to be reminiscent of a zonally symmetric amplitude distribution inducing a zonal wind, as is the paradigmatic picture in parameterized IGW–mean flow interactions. Four cases are considered: 1) A horizontally symmetric IGW field with

*y*direction, with

*α*between the horizontal wavenumber and the horizontal direction of modulation of the wave packet is either 0 (2DWP00) or

*π*/2 (2DWP90). In both of these cases the vertical group velocity is so small that a simulation of a wave packet from initialization to the point where it becomes statically unstable would be computationally very demanding. Since the IGW group velocity is about linearly increasing with the Coriolis parameter, we have therefore also decided to follow in a 4) further case (2DWP90-HAMP-HF) the approach of Lelong and Dunkerton (1998a,b) of considering an atmosphere with a Coriolis parameter an order of magnitude larger. To keep the same ratio

Initial model settings used in the idealized reference cases, including a one-dimensional wave packet case (1DWP), and two two-dimensional wave packet cases with the angle between the initial horizontal wavenumber vector and the plane of modulation of the wave packet amplitude being either 0° (2DWP00) or 90° (2DWP90). The sensitivity experiment (2DWP90-HAMP-HF) is also included.

## 5. Incipient budget analysis

In the case of 1DWP all horizontal derivatives in the various flux convergences vanish. Hence, in the pseudomomentum scheme the vertical gradient of the pseudomomentum flux in the *x*-momentum equation (referred to as PMFC_{xz}) is the only nonzero forcing term. In the direct scheme, the vertical gradient of the momentum flux and the elastic term in the *x*-momentum equation (referred to as MFC_{xz} and ET_{x}, respectively) are the only two nonzero forcing terms. The comparison of the abovementioned three forcing terms (i.e., PMFC_{xz} in the pseudomomentum scheme and MFC_{xz} and ET_{x} in the direct scheme) is given in Fig. 1a. In the case of 1DWP, the signal of MFC_{xz} is apparently stronger than that of PMFC_{xz}. Compared with MFC_{xz}, ET_{x} is relatively small but noticeable.

For further illustration, Fig. 2 shows the dependence of the abovementioned three forcing terms on zonal wavelength (top panel) and vertical wave packet scale _{xz} may be still close to that of MFC_{xz}, and ET_{x} is negligible. As the horizontal wavelength increases, the signal of PMFC_{xz} becomes weaker, while MFC_{xz} is getting stronger instead. A large separation is seen between PMFC_{xz} and MFC_{xz} for waves with relatively long horizontal wavelengths, and ET_{x} becomes more noticeable but still secondary compared with MFC_{xz}. In the sensitivity test to the changes of _{xz} is the same as MFC_{xz}, but the signal of MFC_{xz} is approximately 3 times as strong as that of PMFC_{xz}. The signals of both positive branch (i.e., upper branch) and negative branch (i.e., lower branch) of MFC_{xz} are getting weaker as _{x} is not sensitive to the change of _{x}. Generally speaking, in the current study, ET_{x} appears to be secondary compared with MFC_{xz}. However, with larger _{x} gets more important. Hence, it is not only the IGW aspect ratio between vertical and horizontal wavelength that determines, as described above, the relative magnitude of the dynamical terms in the direct and pseudomomentum approach, but also the IGW amplitude distribution, as here exemplified using the wave packet width.

The comparison among three wave-induced forcing terms (m s^{−2}) in a wide range of one-dimensional wave packet cases: (a),(d) PMFC_{xz} in the pseudomomentum scheme and (b),(e) MFC_{xz} and (c),(f) ET_{x} in the direct scheme. Rows show sensitivity to the change of (a)–(c) zonal wavelengths and (d)–(f) the parameter *σ*_{z} (i.e., vertical wave packet scale). In each sensitivity study, all the other parameters remain the same as in the initial setting of 1DWP. Please also refer to Table 1 for the details of the experiment design in 1DWP.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

The comparison among three wave-induced forcing terms (m s^{−2}) in a wide range of one-dimensional wave packet cases: (a),(d) PMFC_{xz} in the pseudomomentum scheme and (b),(e) MFC_{xz} and (c),(f) ET_{x} in the direct scheme. Rows show sensitivity to the change of (a)–(c) zonal wavelengths and (d)–(f) the parameter *σ*_{z} (i.e., vertical wave packet scale). In each sensitivity study, all the other parameters remain the same as in the initial setting of 1DWP. Please also refer to Table 1 for the details of the experiment design in 1DWP.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

The comparison among three wave-induced forcing terms (m s^{−2}) in a wide range of one-dimensional wave packet cases: (a),(d) PMFC_{xz} in the pseudomomentum scheme and (b),(e) MFC_{xz} and (c),(f) ET_{x} in the direct scheme. Rows show sensitivity to the change of (a)–(c) zonal wavelengths and (d)–(f) the parameter *σ*_{z} (i.e., vertical wave packet scale). In each sensitivity study, all the other parameters remain the same as in the initial setting of 1DWP. Please also refer to Table 1 for the details of the experiment design in 1DWP.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

In the case of 2DWP00, the nonzero IGW-induced forcing terms in the direct approach only include MFC_{yy}, MFC_{yz}, and ET_{y} in the *y*-momentum equation, and there is no forcing in the *x*-momentum equation and thermodynamic equation. Together with the wave-energy density initial vertical cross sections of all nonzero terms are shown in Fig. 3. There is a dipole pattern at the altitude of wave packet center for MFC_{yy}, causing a horizontally asymmetric forcing effect. Instead, the dipole structure in MFC_{yz} results in a vertically asymmetric forcing effect. The ET_{y} has consistent negative signals with minimum at the wave packet center. It is worth mentioning that the forcing from MFC_{yy} is comparable with that from MFC_{yz}, and it is very important in this case. Similar to 1DWP, ET_{y} appears to be secondary but not negligible in this case. To facilitate a comparison between pseudomomentum and direct approach, the corresponding _{yy} and PMFC_{yz} in the pseudomomentum approach, compared to MFC_{yy} and MFC_{yz} in the direct approach. This again suggests that the forcing from MFC_{yy}, which is neglected in operational IGW schemes, is comparable with that from MFC_{yz} in this case.

Initial vertical cross sections of the contributors to the direct approach in the 2DWP00 case (with horizontal wavenumber in the plane of modulation of the wave packet amplitude): (a) MFC_{yy} (m s^{−2}), (b) MFC_{yz} (m s^{−2}), and (c) ET_{y} (m s^{−2}), together with (d) the wave energy per unit volume *E*_{w} (Pa). In (a) and (b), *γ*_{ij} is shown in the top-left corner, indicating how much stronger the momentum flux is as compared to the pseudomomentum flux.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Initial vertical cross sections of the contributors to the direct approach in the 2DWP00 case (with horizontal wavenumber in the plane of modulation of the wave packet amplitude): (a) MFC_{yy} (m s^{−2}), (b) MFC_{yz} (m s^{−2}), and (c) ET_{y} (m s^{−2}), together with (d) the wave energy per unit volume *E*_{w} (Pa). In (a) and (b), *γ*_{ij} is shown in the top-left corner, indicating how much stronger the momentum flux is as compared to the pseudomomentum flux.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Initial vertical cross sections of the contributors to the direct approach in the 2DWP00 case (with horizontal wavenumber in the plane of modulation of the wave packet amplitude): (a) MFC_{yy} (m s^{−2}), (b) MFC_{yz} (m s^{−2}), and (c) ET_{y} (m s^{−2}), together with (d) the wave energy per unit volume *E*_{w} (Pa). In (a) and (b), *γ*_{ij} is shown in the top-left corner, indicating how much stronger the momentum flux is as compared to the pseudomomentum flux.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Case 2DWP90 is interesting for several reasons. Figure 4 shows the initial distribution of all nonzero IGW-induced forcing terms in the direct approach. These include MFC_{yy} in the *y*-momentum equation, MFC_{xz} and ET_{x} in the *x*-momentum equation, and the IGW-induced heating term in the thermodynamic equation. In contrast to these, PMFC_{xz} is the only nonzero term in the pseudomomentum approach; that is, only *x* momentum is forced. Both MFC_{yy} and ET_{x} are due to rotational effects. Among these, only MFC_{yy} and the wave-induced heating term are horizontally asymmetric, while in the pseudomomentum approach no horizontal asymmetry is forced initially.

Initial vertical cross sections of the contributors to the direct approach in the 2DWP90 case (with horizontal wavenumber orthogonal to the plane of modulation of the wave packet amplitude): (a) MFC_{xz} (m s^{−2}), (b) ET_{x} (m s^{−2}), (c) MFC_{yy} (m s^{−2}), and (d) the wave-induced heating (K s^{−1}). In (a) and (c), *γ*_{ij} is shown in the top-left corner, indicating how much stronger the momentum flux is as compared to the pseudomomentum flux.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Initial vertical cross sections of the contributors to the direct approach in the 2DWP90 case (with horizontal wavenumber orthogonal to the plane of modulation of the wave packet amplitude): (a) MFC_{xz} (m s^{−2}), (b) ET_{x} (m s^{−2}), (c) MFC_{yy} (m s^{−2}), and (d) the wave-induced heating (K s^{−1}). In (a) and (c), *γ*_{ij} is shown in the top-left corner, indicating how much stronger the momentum flux is as compared to the pseudomomentum flux.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Initial vertical cross sections of the contributors to the direct approach in the 2DWP90 case (with horizontal wavenumber orthogonal to the plane of modulation of the wave packet amplitude): (a) MFC_{xz} (m s^{−2}), (b) ET_{x} (m s^{−2}), (c) MFC_{yy} (m s^{−2}), and (d) the wave-induced heating (K s^{−1}). In (a) and (c), *γ*_{ij} is shown in the top-left corner, indicating how much stronger the momentum flux is as compared to the pseudomomentum flux.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

## 6. Numerical simulations of different idealized wave packet profiles prior to turbulent dissipation

In this section, the direct and pseudomomentum approach will be validated against wave-resolving simulations with unbalanced large-scale flows. The initial condition settings of the respective numerical experiments are given by the three idealized wave packet profiles described in the previous section (i.e., 1DWP, 2DWP00, and 2DWP90). In each wave-resolving PincFloit LES, there are 16 or ~17 grid points per wavelength of the initial IGW along each available horizontal direction, and 10 grid points per vertical wavelength of the initial IGW. As we have convinced ourselves from 2D simulations with increased resolutions (not shown), these resolutions are sufficient for the laminar phase of the occurring IGW–mean flow interactions, while the later turbulent IGW breaking would require a finer resolution. In each PincFloit/MS-GWaM simulation, the resolution along each available direction is close to the wavelength of the initial IGW in the corresponding direction, and the large-scale flow induced by the parameterized IGWs can still be resolved in the coarse-resolution PincFloit/MS-GWaM simulation. In the PincFloit LESs, the wave packets of the resolved IGWs are initialized in two-dimensional domains for 1DWP and 2DWP00, and three-dimensional domain settings are employed for 2DWP90. Instead, in the PincFloit/MS-GWaM simulations, 1DWP uses a single vertical column, and two-dimensional domain settings are employed for all two-dimensional wave packet cases. The details of the model designs are given in Tables 1 and 2.

Synopsis of the relevant general model parameters in both PincFloit/MS-GWaM and PincFloit LES. In all the simulations, periodic boundaries are assumed in each available horizontal direction, while rigid boundaries are assumed in the vertical direction. The time step Δ*t* is determined through the CFL criterion (CFL = 0.5), while a 1-s upper threshold for the time step is used (i.e., Δ*t*_{max} = 1 s). The smoothing parameter indicates the total number of grid cells used for a local smoothing along all the available spatial directions. In PincFloit/MS-GWaM, the initial total number of ray volumes *n*_{ray} corresponds to the product of the number of grid cells in the 5*σ* segment or box, and the corresponding number per grid cell and spatial direction (i.e.,

Figure 5 compares the large-scale *x*-velocity component and the large-scale *y*-velocity component between the PincFloit LES, and the PincFloit/MS-GWaM simulations using either direct or pseudomomentum approach. Similar to the procedure in Bölöni et al. (2016) and Wilhelm et al. (2018), the large-scale wind component in PincFloit LES is obtained by a running mean over two wavelengths of the initial IGW. The selected time for this comparison in Fig. 5 is at *t* = 1500 min, which is approximately 1.5 inertial periods. Since the wave energy of the 1DWP case does not depend on the horizontal direction, the induced large-scale zonal velocity component and meridional velocity component do not change along the horizontal direction, and the large-scale vertical velocity component is zero everywhere in the domain. As depicted in Fig. 5, the direct scheme is able to reproduce the signals in the data from the wave-resolving simulation with very small errors. The pseudomomentum scheme, on the contrary, gives a much weaker large-scale horizontal wind compared with the wave-resolving data, and thus large model errors could be expected.

Comparison of two horizontal mean wind components (m s^{−1}) among three model codes at *t* = 1500 min (approximately 1.5 inertial periods) in the 1DWP case. The three codes include a high-resolution wave-resolving simulation (black squares) and two coarse-resolution simulations with gravity wave parameterization based on the direct or pseudomomentum approach (blue and red squares, respectively). The two mean wind components include (a) zonal wind component and (b) meridional wind component. The mean vertical wind is zero in all simulations. Please refer to Tables 1 and 2 for the details of the simulations.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Comparison of two horizontal mean wind components (m s^{−1}) among three model codes at *t* = 1500 min (approximately 1.5 inertial periods) in the 1DWP case. The three codes include a high-resolution wave-resolving simulation (black squares) and two coarse-resolution simulations with gravity wave parameterization based on the direct or pseudomomentum approach (blue and red squares, respectively). The two mean wind components include (a) zonal wind component and (b) meridional wind component. The mean vertical wind is zero in all simulations. Please refer to Tables 1 and 2 for the details of the simulations.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Comparison of two horizontal mean wind components (m s^{−1}) among three model codes at *t* = 1500 min (approximately 1.5 inertial periods) in the 1DWP case. The three codes include a high-resolution wave-resolving simulation (black squares) and two coarse-resolution simulations with gravity wave parameterization based on the direct or pseudomomentum approach (blue and red squares, respectively). The two mean wind components include (a) zonal wind component and (b) meridional wind component. The mean vertical wind is zero in all simulations. Please refer to Tables 1 and 2 for the details of the simulations.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Figure 6 shows corresponding results, again at *t* = 1500 min, but now for case 2DWP00. For the two-dimensional wave packets in the current study, in addition to the induced large-scale horizontal wind on both directions, the large-scale vertical velocity component is nonzero. Again, the direct scheme successfully captures the signals in the data from the wave-resolving simulation. Even though the pattern from the results based on the pseudomomentum scheme is still similar to the wave-resolving simulation for all the three wind components, its signals are much weaker. Note that the fact that the pseudomomentum scheme generates much weaker large-scale wind components is consistent with the budget analysis shown in the previous section.

The comparison of all three mean wind components (m s^{−1}) among simulations using the same codes as in Fig. 5, again at *t* = 1500 min (approximately 1.5 inertial periods), but now for the 2DWP00 case. Results are from (a)–(c) the high-resolution wave-resolving simulation, (d)–(f) the coarse-resolution PincFloit/MS-GWaM simulations using the direct scheme, and (g)–(i) the pseudomomentum scheme farther below. The three mean wind components include the horizontal (left) *x*- and (center) *y*-wind components and (right) the vertical wind component. Please refer to Tables 1 and 2 for the details of the simulations.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

The comparison of all three mean wind components (m s^{−1}) among simulations using the same codes as in Fig. 5, again at *t* = 1500 min (approximately 1.5 inertial periods), but now for the 2DWP00 case. Results are from (a)–(c) the high-resolution wave-resolving simulation, (d)–(f) the coarse-resolution PincFloit/MS-GWaM simulations using the direct scheme, and (g)–(i) the pseudomomentum scheme farther below. The three mean wind components include the horizontal (left) *x*- and (center) *y*-wind components and (right) the vertical wind component. Please refer to Tables 1 and 2 for the details of the simulations.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

The comparison of all three mean wind components (m s^{−1}) among simulations using the same codes as in Fig. 5, again at *t* = 1500 min (approximately 1.5 inertial periods), but now for the 2DWP00 case. Results are from (a)–(c) the high-resolution wave-resolving simulation, (d)–(f) the coarse-resolution PincFloit/MS-GWaM simulations using the direct scheme, and (g)–(i) the pseudomomentum scheme farther below. The three mean wind components include the horizontal (left) *x*- and (center) *y*-wind components and (right) the vertical wind component. Please refer to Tables 1 and 2 for the details of the simulations.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Figure 7 shows corresponding results for the 2DWP90 case. It again demonstrates that the direct scheme is more reliable than the pseudomomentum scheme, with the data from the wave-resolving simulation as a reference. Two further things are worth noting here. 1) Arguably the best agreement between pseudomomentum approach and wave-resolving simulation is visible in the *x*-wind component. This could be expected since this is the horizontal wind component that can also contain balanced contributions, given by the *y* derivative of the balanced part of the pressure. The other side of the coin is, however, that the unbalanced response in the meridional circulation is met well at best in structure but definitely not in strength. 2) All three large-scale wind components produced by the pseudomomentum scheme for this particular case are highly horizontally symmetrical, which is quite different from the results from the direct scheme and the wave-resolving data. This suggests that the horizontally asymmetrical forcing is apparently negligible in the pseudomomentum scheme in this case. As suggested in Fig. 4 from the previous section, this is again consistent with the budget analysis in the pseudomomentum scheme for the 2DWP90 case. Indeed here *l* = 0 and hence PMF_{yy} = 0, while MF_{yy} is nonzero and explains the asymmetry of the response.

As in Fig. 6, but for the 2DWP90 case.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

As in Fig. 6, but for the 2DWP90 case.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

As in Fig. 6, but for the 2DWP90 case.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

## 7. Sensitivity experiments

Various sensitivity tests have been done. In the first of these we have convinced ourselves that our findings even hold for IGW amplitudes very close to overturning instability. To this purpose we have considered a high-amplitude two-dimensional wave packet case (2DWP90-HAMP). Here, the initial setting is the same as that in 2DWP90, except that the initial wave amplitude at the wave packet center reaches the limit of convective instability. Therefore, a stronger wave–mean flow interaction is expected. To be compared to Fig. 7 for 2DWP90, Fig. 8 shows the comparisons of the large-scale mean wind among three model codes at the same time for 2DWP90-HAMP (see also Table 3). Similar large-scale wind patterns are found in cases 2DWP90 (Fig. 7) and 2DWP90-HAMP (Fig. 8), except that the signals in 2DWP90-HAMP are approximately 4 times as strong as those in 2DWP90. With a stronger wave–mean flow interaction, this case continues to support the statement that the direct approach is generally more reliable than the pseudomomentum approach.

As in Fig. 6, but for the 2DWP90-HAMP case. Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

As in Fig. 6, but for the 2DWP90-HAMP case. Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

As in Fig. 6, but for the 2DWP90-HAMP case. Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Initial model settings used in the sensitivity experiment.

Next, expanding on the budget analysis in section 5, we have investigated the relative importance of the elastic effect and the heating term for the results from the direct approach. Figure 9 shows results from three sensitivity experiments, where either the elastic term has been set to zero, or the heating term, or both. These are to be compared to the results from the direct scheme for the 2DWP90-HAMP case in Fig. 8. These sensitivity experiments are denoted by 2DWP90-HAMP-NoET, 2DWP90-HAMP-NoHeat, and 2DWP90-HAMP-NoET-NoHeat, respectively. With PincFloit LES as a reference, all of the abovementioned sensitivity experiments demonstrate a much stronger negative large-scale zonal mean wind at the domain center, especially when the heating effect is switched off. This implies that it is important to take into account the heating term in order to get the correct large-scale flow response. The elastic term, however, is, at least in this study, noticeable but still of secondary importance.

As in the results from the direct scheme in Fig. 8, but for the sensitivity experiments of (a)–(c) 2DWP90-HAMP-NoET, (d)–(f) 2DWP90-HAMP-NoHeat, and (g)–(i) 2DWP90-HAMP-NoET-NoHeat. In (a)–(c) the elastic term has been ignored in the simulations. In (d)–(f) the wave-induced heating term has been ignored, and in (g)–(i) both elastic term and heating term have been neglected. Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

As in the results from the direct scheme in Fig. 8, but for the sensitivity experiments of (a)–(c) 2DWP90-HAMP-NoET, (d)–(f) 2DWP90-HAMP-NoHeat, and (g)–(i) 2DWP90-HAMP-NoET-NoHeat. In (a)–(c) the elastic term has been ignored in the simulations. In (d)–(f) the wave-induced heating term has been ignored, and in (g)–(i) both elastic term and heating term have been neglected. Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

As in the results from the direct scheme in Fig. 8, but for the sensitivity experiments of (a)–(c) 2DWP90-HAMP-NoET, (d)–(f) 2DWP90-HAMP-NoHeat, and (g)–(i) 2DWP90-HAMP-NoET-NoHeat. In (a)–(c) the elastic term has been ignored in the simulations. In (d)–(f) the wave-induced heating term has been ignored, and in (g)–(i) both elastic term and heating term have been neglected. Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

In all of the cases considered so far, the vertical group velocity is so small that a simulation of a wave packet from initialization to the point where it becomes statically unstable by propagation in increasingly attenuated altitudes, hence growing in amplitude, would be computationally very demanding. Since the IGW group velocity is about linearly increasing with the Coriolis parameter we have therefore also decided to follow in a further case (2DWP90-HAMP-HF) the approach of Lelong and Dunkerton (1998a,b) of considering an atmosphere with a Coriolis parameter an order of magnitude larger. To keep the same ratio *x* wind, with its balanced components, is relatively reasonable in the simulations with the pseudomomentum scheme, the direct scheme is the only one able to reproduce the response of the *y*–*z* circulation in its full strength. It is also able to match the horizontal asymmetry of the response, while the pseudomomentum approach misses this feature.

As in Fig. 6, but for the 2DWP90-HAMP-HF case at *t* = 471.15 min (approximately 4.5 inertial periods). Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

As in Fig. 6, but for the 2DWP90-HAMP-HF case at *t* = 471.15 min (approximately 4.5 inertial periods). Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

As in Fig. 6, but for the 2DWP90-HAMP-HF case at *t* = 471.15 min (approximately 4.5 inertial periods). Please refer to Table 3 for the details of the sensitivity experiments.

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

Finally, Fig. 11 illustrates the importance of the wave–mean flow interaction, as well as the role of the saturation scheme, by comparing for the same case 2DWP90-HAMP-HF the large-scale *y*–*z* circulation and the wave energy among two sensitivity experiments and their corresponding control simulation, all using PincFloit/MS-GWaM with the direct approach, but at a much later stage, where we do not have PincFloit LES data available. Here, in the first sensitivity experiment the coupling process between the large-scale mean wind and the gravity waves has been turned off by assuming that the large-scale mean wind has no impact on changing the wavenumber of the gravity waves via the ray-tracing equation. In the second sensitivity experiment the saturation scheme has been turned off by assuming that the wave action density associated with each ray is always conserved during its vertical propagation. Differences are easily identified between each abovementioned sensitivity experiment and its corresponding control simulation. The findings of Bölöni et al. (2016) are confirmed that the direct wave–mean flow interaction dominates the dynamics to leading order, while turbulent wave breaking leads to next-order modifications.

(a)–(c) As in the PincFloit/MS-GWaM simulation in Fig. 10, but at a much later stage (*t* = 1832.25 min, approximately 17.5 inertial periods after initialization) and compared to results from (d)–(f) a corresponding simulation where the mean-flow impact on the waves has been switched off (middle row), and (g)–(i) a simulation where the saturation scheme, describing the effect of turbulent wave breaking, has been put out of use. The three compared variables include the (left) large-scale mean *y* wind (m s^{−1}), (center) large-scale mean vertical wind (m s^{−1}), and (right) wave energy ^{2} s^{−2}).

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

(a)–(c) As in the PincFloit/MS-GWaM simulation in Fig. 10, but at a much later stage (*t* = 1832.25 min, approximately 17.5 inertial periods after initialization) and compared to results from (d)–(f) a corresponding simulation where the mean-flow impact on the waves has been switched off (middle row), and (g)–(i) a simulation where the saturation scheme, describing the effect of turbulent wave breaking, has been put out of use. The three compared variables include the (left) large-scale mean *y* wind (m s^{−1}), (center) large-scale mean vertical wind (m s^{−1}), and (right) wave energy ^{2} s^{−2}).

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

(a)–(c) As in the PincFloit/MS-GWaM simulation in Fig. 10, but at a much later stage (*t* = 1832.25 min, approximately 17.5 inertial periods after initialization) and compared to results from (d)–(f) a corresponding simulation where the mean-flow impact on the waves has been switched off (middle row), and (g)–(i) a simulation where the saturation scheme, describing the effect of turbulent wave breaking, has been put out of use. The three compared variables include the (left) large-scale mean *y* wind (m s^{−1}), (center) large-scale mean vertical wind (m s^{−1}), and (right) wave energy ^{2} s^{−2}).

Citation: Journal of the Atmospheric Sciences 76, 9; 10.1175/JAS-D-18-0337.1

## 8. Concluding remarks and discussion

This article investigates and compares two different approaches for the efficient modeling of subgrid-scale inertia–gravity waves (IGWs) in a rotating compressible atmosphere. The first approach, denoted as a pseudomomentum scheme, is closely related to the fundamental result of Andrews and McIntyre (1978) that in a Lagrangian-mean reference frame GW effects on the large-scale flow only occur in the momentum equation. It can be shown that this holds also in the Eulerian-mean reference frame used by atmospheric models, at least if the large-scale flow is in hydrostatic and geostrophic balance. The GW forcing can then be expressed by a pseudomomentum-flux convergence in the large-scale momentum equation. Present-day gravity wave parameterizations follow this route, exclusively applying a corresponding forcing, and leaving out elastic and thermal effects that may arise in the absence of large-scale-flow balance. In the second approach, called a direct scheme, the large-scale flow is forced both in the momentum equation by anelastic momentum-flux convergence and in the entropy equation via entropy-flux convergence. In addition, also an elastic term is taken into account in the momentum equation. This second approach does not rely on any balance assumption with regard to the large-scale flow, and it differs from the pseudomomentum approach wherever rotational effects matter, that is, outside of the tropics and for low-frequency IGWs.

It has been the purpose of the current study to understand the difference between the pseudomomentum and the direct approaches. To this end wave-resolving simulations have been done, to provide reference data against which to validate coarse-resolution model simulations with either of the two approaches providing the basis for a prognostic gravity wave parameterization. All simulations have been done with the code PincFloit, solving the pseudoincompressible equations by a finite-volume method. At high resolutions PincFloit has been used together with a dynamical scheme for the parameterization of the largest turbulent eddies (PincFloit LES). In the coarse-resolution simulations it has been coupled to the prognostic gravity wave model MS-GWaM, solving the coupled WKB equations using a Lagrangian approach. This model has been used in either pseudomomentum mode or direct mode.

The dynamics of 1D IGW packets, with an amplitude only depending on time and the vertical direction, already shows that the comparative performance of the two approaches is sensitive to the following two parameters: 1) the intrinsic frequency and 2) the wave packet scale. The smaller the intrinsic frequency is, the larger the discrepancy between pseudomomentum-flux convergence and momentum-flux convergence is, and, hence, the greater the differences between two schemes are. Pseudomomentum-flux convergence is then much weaker than momentum-flux convergence. The elastic term is also nonzero in that case, but it generally appears to be of comparatively smaller importance, and only begins exhibiting some effect if the wave packet scale is rather large.

Further 2D case studies, with IGW envelope depending on time and *y* and *z*, meant to be reminiscent of a zonally symmetric IGW distribution in interaction with a zonally symmetric large-scale flow, confirm the expectations from the 1D analyses. They show that only the direct approach is able to capture the response in the unbalanced components of the large-scale flow. This response is strongly underestimated by the pseudomomentum approach. It will be interesting to investigate, for example, what consequences this has for the correct description of the IGW impact on the residual circulation. Even in the balanced response, however, some structural asymmetries can only be captured by the direct approach, while the more symmetric pseudomomentum-flux convergence is not able to do so. In these 2D cases it is especially the horizontal entropy-flux convergence that seems to matter, while the elastic term does not seem to be that important. It should be stressed that these differences are only relevant for low-frequency IGWs, and outside the tropics, while for high-frequency GWs and in the tropics pseudomomentum and direct approach are approximately equivalent. Finally, in the present settings we reconfirm the findings of Bölöni et al. (2016) that the direct wave–mean flow interaction takes a leading role, as compared to turbulence effects.

In summary, our results show that the direct approach is more reliable than its pseudomomentum counterpart. In some cases, the pseudomomentum approach yields a balanced response that is of the right strength, however, not quite with the right structure. Even more conspicuous, however, appears the difference in the reproduction of the potentially most unbalanced part of the response. The meridional-circulation response from the pseudomomentum scheme is generally too weak. As the direct scheme is not much more expensive than the pseudomomentum scheme, it seems advisable to use it instead for GW parameterizations in atmospheric models, at least outside the tropics. Nonetheless, our analyses have not yet considered global applications where irreversible GW effects due to wave breaking will weigh in more strongly. For these the comparison between the two approaches, beyond the scope of the present study, is still to be done. Our results give considerable motivation for this next step.

## Acknowledgments

We benefited from the insightful comments from two anonymous reviewers on an earlier version of the manuscript. The authors also thank the German Research Foundation (DFG) for partial support through the research unit Multiscale Dynamics of Gravity Waves (MS-GWaves) and through Grants AC 71/8-2, AC 71/9-2, AC 71/10-1, AC 71/10-2, AC 71/11-2, AC 71/12-2, BO 5071/2-2, and BO 5071/1-2. Calculations for this research were conducted on the LOEWE-CSC high-performance computer of the Goethe Universität Frankfurt.

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