## 1. Introduction

Internal gravity waves (IGWs) play a significant role in distributing energy and momentum throughout the atmosphere. Being generated through the perturbation of balanced flow states in stratified environments (e.g., convection, flow over topography, flow instabilities, and wave turbulence), they may propagate over large distances and deposit their energy, generate turbulence, and modify the mean flow dynamics in general far away from their source (Fritts and Alexander 2003; Alexander et al. 2010; Sutherland 2010; Nappo 2013; Williams et al. 2017; Sutherland et al. 2019; Achatz 2022). While propagating, they may exchange energy among themselves, transiently interact with mean flows, or influence the transport of chemical species. Even though their generation regions are mostly located in the troposphere, their effects are strongest in the middle atmosphere (Lindzen 1981; Kim et al. 2003; Sigmond and Scinocca 2010). However, these major effects may couple to and thus impact, for instance, tropospheric weather patterns or climate conditions (Scaife et al. 2005, 2012; Sigmond and Scinocca 2010).

In state-of-the-art general circulation models (GCMs) or numerical weather prediction (NWP) models, gravity waves are only partially resolved and thus need parameterization (e.g., Kim et al. 2003; Holt et al. 2016). Even when entering the global kilometer-resolving regime, their effects are not entirely represented (Kruse et al. 2022; Polichtchouk et al. 2022, 2023). IGW parameterizations commonly rely on WKBJ theories (Bretherton 1966; Grimshaw 1975; Achatz et al. 2017), incorporating some major simplifications (Lindzen 1981; Medvedev and Klaassen 1995; Warner and McIntyre 1996; Hines 1997b,a; Lott and Miller 1997; Alexander and Dunkerton 1999; Scinocca 2003; Orr et al. 2010; Lott and Guez 2013). Most notably, there are three assumptions that are usually made and shall be considered in this work. First, local horizontal homogeneity is assumed in the so-called single-column approximation, i.e., horizontal gradients of the resolved flow are neglected, which leads horizontal wavenumbers to be invariant during propagation. Furthermore, responses of the resolved flow due to the horizontal finiteness of an IGW field are neglected. Second, a steady-state assumption is adopted such that IGWs instantly propagate through the atmosphere, leading to a neglect of all transient propagation effects. Third, it is commonly assumed that the resolved flow is (approximately) in hydrostatic and geostrophic balance, implying a reduced formula for the resolved flow response to IGWs. In the extratropics, this can lead to significant modifications in the IGW forcing when the flow is imbalanced.

Recent investigations suggested that the mentioned effects can have important impacts on modeled flows (Sato et al. 2009; Bölöni et al. 2016; Ehard et al. 2017; Wei et al. 2019). In particular, Bölöni et al. (2016) showed that including the time dependence of gravity wave propagation significantly improves the estimated wave drag in idealized simulations. Sato et al. (2009) found that three-dimensional propagation pathways can impact the structure of the stratosphere and mesosphere using wave-resolving general circulation models. Similarly, Ehard et al. (2017) highlight the importance of horizontal gravity wave propagation for the dynamics of the Antarctic polar night jet by analyzing various observations and reanalyses. Finally, Wei et al. (2019) showed that describing the wave momentum fluxes with a direct formulation can improve the simulated wave impact on the resolved flow. Several studies have successfully relaxed some of the simplifications in model parameterizations (e.g., Muraschko et al. 2015; Wilhelm et al. 2018; Quinn et al. 2020). In particular, Bölöni et al. (2021) and Kim et al. (2021) presented a novel Lagrangian IGW parameterization, the Multiscale Gravity Wave Model (MS-GWaM). It is built on a weakly nonlinear WKBJ theory, including transient wave–mean flow interactions. The model has been implemented in a single-column mode into a state-of-the-art weather forecast and climate code. It will therefore be referred to as MS-GWaM-1D throughout this manuscript. Here, we present MS-GWaM-3D, an extension of MS-GWaM-1D, which models the full 3D transient propagation of IGWs and their forcing of the general (cf. balanced) resolved flow. While both the columnar approximation and assumptions on the resolved flow are relaxed in MS-GWaM-3D, we shall focus on the effects of horizontal propagation in this study. In particular, the modified refraction behavior and resolved-flow response in the statistical mean are investigated. Indeed, we find that many aspects show an increased realism of the wave dynamics. However, our results also underline the challenges left for future studies. Ultimately, we aim to improve the representation of the parameterized IGW processes while keeping simulations numerically efficient.

This paper is structured as follows. First, a brief recapitulation of the underlying WKBJ theory for transient, three-dimensional IGWs (section 2) is presented. It is then followed by the description of the employed ray-tracing techniques (section 3). Last, results from simulations with MS-GWaM-3D (section 4) are visualized and discussed. The manuscript then closes with some concluding remarks on the achievements and challenges of ray-tracing parameterization (section 5).

## 2. Nonlinear, three-dimensional, and transient internal gravity waves

Although generally three-dimensional and transient, internal gravity waves are commonly parameterized using both single-column and steady-state approximations. Here, we attempt to relax both these assumptions and build a three-dimensional and transient parameterization. An underlying multiscale theory has been presented by Achatz et al. (2010, 2017, 2023), and thus, we restrict ourselves to summarizing the results here for brevity.

### a. Wave evolution

**r**=

*λ*

**e**

_{λ}ϕ**e**

*+*

_{ϕ}*r*

**e**

*, and the wavevector,*

_{r}**k**=

*k*

_{λ}**e**

*+*

_{λ}*k*

_{ϕ}**e**

*+*

_{ϕ}*k*

_{r}**e**

*. Here,*

_{r}*λ*,

*ϕ,*and

*r*are the standard geographical spherical coordinates and

**e**

*,*

_{λ}**e**

*, and*

_{ϕ}**e**

*are the corresponding unit vectors. Note that the mean-flow velocity*

_{r}**U**(

**r**,

*t*) =

*U*

**e**

*+*

_{λ}*V*

**e**

*, the Coriolis frequency*

_{ϕ}*f*=

*f*(

**r**), and the buoyancy frequency

*N*=

*N*(

**r**,

*t*) are functions of space and time as indicated. For convenience, we have defined the total wavevector squared as

*K*

^{2}=

**k**⋅

**k**+ Γ

^{2}, with the scale height correction Γ = Γ(

**r**,

*t*). Finally, we would like the reader to note that the theory is written in spherical coordinates. The corresponding eikonal equations and the group velocities

**c**

*for an individual wave component then follow the relations:*

_{g}*and ∇*

_{r}*, respectively. Having the application to a ray-tracing scheme in mind, we indicate by a dot the derivative along characteristics, so-called rays, with their positions denoted by*

_{k}**r**(

*t*) so that

*k*,

_{λ}*k*,

_{ϕ}*k*) in spherical coordinates read

_{r}^{1}Note that the representation in spherical coordinates comes with the drawback of a pole problem in the above relations when propagating very close to the pole. Practically, singularities due to division by zero, however, do not pose a serious problem as they only occur in close vicinity of the poles. To avoid potential problems, we do not consider wave generation where |

*ϕ*| > 85° (see section 3c).

*S*denotes the sources and sinks of wave action, such as the wave saturation as described below. Wave–wave interactions between IGWs and interactions between IGWs and the (subgrid-scale) geostrophic modes (GM) are neglected, and thus, the wave action conservation is valid for any overlapping wave fields, provided their amplitudes are sufficiently weak. Numerous processes, e.g., wave refraction, can lead to rays crossing in physical space at so-called caustics, so that the assumption of local monochromaticity can break down in the course of the integration of the eikonal equations above. To avoid numerical instabilities due to this issue, we introduce the spectral wave action density

*j*indicates any member of a possibly infinitely large and infinitely dense set of locally monochromatic fields that are being superposed. The resulting phase-space wave action conservation then reads

### b. Wave impact on the mean flow

**e**

*is the radial unit vector,*

_{r}*g*is Earth’s gravity, and

**v**′,

*b*′, and

*θ*′ denote the IGW velocity, buoyancy, and potential temperature perturbations and

**u**′ denotes the horizontal component of

**v**′. The brackets 〈⋅〉 represent the phase average of the wave perturbations. Using the dispersion and polarization relations, they may be expressed as

**k**

*=*

_{h}*k*

_{λ}**e**

*+*

_{λ}*k*

_{ϕ}**e**

*and where*

_{ϕ}**u**′

*θ*′〉 are negligible where the Coriolis frequency

*f*is small with respect to the intrinsic wave frequency

Combined with the eikonal equations [Eqs. (4)–(6)], wave propagation [Eq. (3)], and wave action conservation [Eq. (8)], these relations form a closed prognostic system for transient wave propagation. Note that the equations are energy-conserving wherever both wave sources and dissipation are zero (not shown explicitly).

## 3. Gravity wave ray tracing

As pointed out in section 2, the subgrid-scale gravity wave dynamics are natively described on the rays of the corresponding gravity wave propagation. To include both the transience and the horizontal wave propagation, one may employ various techniques to obtain a numerical solution. Ray tracing has gained significant attention recently as an efficient and accurate method for simulating wave fields (Marks and Eckermann 1995; Muraschko et al. 2015; Amemiya and Sato 2016; Voelker et al. 2021; Bölöni et al. 2021). Here, we build on the implementations of Bölöni et al. (2021) and Kim et al. (2021) and extend the approach by the horizontal wave propagation and the full flux calculation [Eqs. (10) and (11)]. While we present a broad overview in this section, we would like to refer the reader to appendix A for additional details.

### a. Representation of rays as phase-space ray volumes

*x*,

*y*,

*z*) are the Cartesian coordinates and (

*k*,

_{x}*k*,

_{y}*k*) are the wave vector components in these coordinates. In Cartesian coordinates, one has even

_{z}*λ*,

*ϕ*,

*r*,

*k*,

_{λ}*k*,

_{ϕ}*k*). Within the spanned ray volume, the wave action density

_{r}*N*is assumed to be constant. The ray-volume content is then propagated according to Eqs. (3)–(6) following a central carrier ray. The change in physical extent Δ(

*λ*,

*ϕ*,

*r*) is determined using rays at the faces of the ray volumes carrying identical wave properties (

*k*,

_{λ}*k*,

_{ϕ}*k*) as in the central carrier ray but being exposed to deviations in the background fields. In particular, the ray-volume extent is approximated as locally Cartesian by

_{r}*k*,

_{x}*k*,

_{y}*k*) is determined by Eq. (19) and then inverted using Eq. (21) to finally obtain the new Δ(

_{z}*k*,

_{λ}*k*,

_{ϕ}*k*). Note that this procedure, albeit converging for infinitesimally small ray volumes, implies the following simplification. In a uniform atmosphere at rest, where the group velocity differences between the opposing ray-volume faces are exactly zero, the ray-volume extent in spherical coordinates Δ(

_{r}*λ*,

*ϕ*,

*r*) is exactly conserved. This implies, however, that any ray volume propagating poleward must shrink in the tangent-linear extent Δ

*x*and expand in spectral width Δ

*k*(and vice versa for southward propagation). Although this effect may be small in regions sufficiently separated from the pole, we plan to apply an advanced method to remove this potential problem in the next version of MS-GWaM.

_{λ}### b. Coupling of Lagrangian particles to the Eulerian mean flow

The coupling between the large-scale flow and the wave perturbations necessitates linking the Lagrangian wave representation to the Eulerian model grid. For the evaluation of the modulation equations, we linearly interpolate all necessary mean fields and their gradients to the ray-volume center or faces using a first-order Taylor expansion. In particular, the group velocities are calculated at the physical boundaries of the ray volume (see section a in appendix B). This serves a dual purpose: The mean of the group velocities on opposing ray-volume faces may be used to advance the location of the central carrier ray, and the difference between them serves as an estimate for the compression or inflation of the ray volume in physical space. The phase-space extent is then determined as described above.

To account for the impact of waves on the mean flow, it is necessary to differentiate between vertical and horizontal components, considering possibly irregular horizontal grid structures. The vertical gradient of the wave flux, i.e.,

### c. Sources and sinks of internal gravity waves

*F*= 8 mPa, no fluxes for latitudes

_{w}*ϕ*∈ [20°S, 20°N], and summer hemispheric fluxes of

*F*= 2.4 mPa. No background waves are launched at latitudes higher than |

_{s}*ϕ*| > 85°. The different regimes are connected through smooth transitions. The profile oscillates in time with a sinusoidal yearly cycle. Summarizing, we employ a latitudinal profile

*α*(

*φ*) and a total absolute momentum flux at the lower boundary during the northern summer solstice

*F*

_{sol},

*φ*is the latitude (°N). The absolute momentum flux at the lower boundary condition then reads

*τ*= 2

*π*(

*t*−

*t*

_{sol})/1

*y*, relative to the time of the northern summer solstice

*t*

_{sol}. These background waves are launched at every time step at a height corresponding to a pressure of 300 hPa (black, dotted line in Figs. 1a,c) with a momentum flux which is equally divided into four horizontal directions as done by Bölöni et al. (2021). In particular, we construct a ghost layer at the launch altitude within which ray volumes are allowed to propagate vertically only. At every launch interval, that is, every 60 s (equivalent to the model time step), the fraction of the ray volume which has propagated out of the ghost layer is separated and activated. Only then, it will propagate freely according to the modulation equations. Ray volumes which are reflected within the ghost layer are discarded. More technical information on the launch process can be found in the work of Bölöni et al. (2021). This background wave energy source is directionally homogeneous, with a purely latitudinal profile on the 300-hPa pressure surface. Additionally, convectively generated waves are considered with a launch spectrum as described by Kim et al. (2021). In their formulation, the areal fraction of the convective latent heat release

*ϵ*inversely controls the total entropy forcing and thus the launch momentum flux (Kim et al. 2021). Here, we increase the area fraction to

_{q}*ε*= 7%, resulting in a reduced-gravity wave flux. For the convective source, we couple MS-GWaM to the convection parameterization and utilize the convection strength and depth calculated within. The ray volumes are then launched at the cloud top and under consideration of the background wind. As an example of the resulting lower boundary conditions, we show the absolute vertical momentum fluxes, defined by

_{q}Finally, we would like to add that the launch fluxes *F _{w}* and

*F*, as well as the area fraction

_{s}*ϵ*, were tuned to achieve a realistic zonal mean mesopause. In particular, both the wind reversal and the summer hemispheric cold pole are expected to occur at approximately 80 km (see discussion of mean model states in section 4a). To avoid high computational costs due to optimization algorithms, short runs were considered, integrating the model for 2 weeks in both seasons. The model results, calculated for a physical range of the three tunable parameters, were then compared, and the best fit was chosen. The resulting directional momentum fluxes compare in order of magnitude with satellite observations at altitudes 30–40 km but differ in horizontal distribution (Jiang et al. 2006; Ern et al. 2018; Hindley et al. 2020; Polichtchouk et al. 2022; direct comparisons not shown). The latter is, however, expected given that MS-GWaM incorporates a flow-independent background source.

_{q}*B*being the amplitude of IGW buoyancy. The concept of the integrated density perturbation

_{w}*j*at the location

**r**

*such that*

_{b}*α*are the ray volume’s total wave action and the spatial fraction that the

_{j}*j*th ray volume takes of the volume of the cell it is contained in, respectively. Thus, rather than each individual spectral component, we consider a spectrally integrated saturation. The reduction in the individual wave action components

*t*→ ∞. Because at the beginning of each time step

*t*, so that

*κ*always ends up finite. This spectral saturation implementation was shown to compare well to wave-resolving simulations in idealized cases by Bölöni et al. (2016). Furthermore, it is planned to couple the saturation scheme to the turbulence parameterization of the underlying atmospheric model, Icosahedral Nonhydrostatic (ICON). An in-depth analysis of the breaking processes and the impact of the spectral saturation on a global scale is hence left to a future study.

## 4. Atmospheric simulations with UA-ICON and MS-GWaM-3D

Building on Bölöni et al. (2021) and Kim et al. (2021), who introduced MS-GWaM-1D, we choose to work with the ICON model with its upper-atmosphere extension (Zängl et al. 2015; Borchert et al. 2019). Here, we use version 2.6.5*-*nwp1b with a horizontal resolution of approximately 160 km (model grid R2B04) and the physics packages for the NWP and the upper atmosphere. Note that MS-GWaM replaces the nonorographic gravity wave drag but leaves the operational orographic wave parameterization intact (in particular, Lott and Miller 1997). Through the coupling to the mean wind, the latter thus indirectly interacts with MS-GWaM. The setup has a model top of 150 km with a vertical grid extent of a few 10 m in the boundary layer, 700–1500 m in the stratosphere, and a maximum of approximately 4 km in the lower thermosphere. A sponge layer acts above an altitude of 110 km, which is why we restrict our analysis to altitudes below 100 km. The model is initialized with IFS analysis data below 60 km and with the climatological thermodynamic state at rest above. It is then spun up for a month to exclude adjustment effects in our analysis. These runs are repeated to simulate June and December for the years 1991 through 1998. Additionally, we deploy runs for June and December 1991 with extended diagnostics. In particular, the free parameters controlling the launch fluxes are the convective area fraction for the convectively generated waves and the summer and winter launch amplitudes of the background waves (see section 3c). In general, stronger gravity wave launch fluxes shift the wave saturation to lower altitudes, leading to a lower wind reversal at the mesopause. Comparisons of simulations with the horizontal wind model (HWM) (HWM2014; Drob et al. 2015) and the empirical temperature model NRLMSIS2.1 (MSIS; Emmert et al. 2022) then give a good estimate for the performance of the chosen parameters. The HWM2014 and MSIS are chosen based on their wealth of incorporated data in the middle and upper atmospheres in the time range of our simulations. In particular, they incorporate reanalysis, ground-based datasets, and satellite-based observations (for more details, see Drob et al. 2015; Emmert et al. 2021, 2022). The reference climatologies are then generated for exactly the simulated months. With the help of short simulations spanning a large parameter space, we identified an optimum with an area fraction of *ε _{q}* = 7%, a winter launch flux of

*F*= 8 mPa, and a summer launch flux of

_{w}*F*= 2.4 mPa (for more details, see section 3c).

_{s}### a. Zonal mean flow

To test the general performance of the gravity wave parameterization, we analyze the zonal-mean zonal wind and temperatures as composites over eight simulations of June and December (Figs. 2 and 3). To highlight the general effect of the parameterizations and visualize the general model performance, we accompany the data from simulations without nonorographic gravity wave parameterization. Naturally, the runs including a gravity wave drag parameterization mend model biases and thus differ significantly from the runs without. The comparison is, therefore, to be understood as a reference for the general behavior of ICON in the present setup and in how far model biases stem from the lack of a gravity wave drag parameterization or may be rooted elsewhere. Additionally, we show the reference climatologies HWM2014 and MSIS. Note that, as described above, both versions of MS-GWaM were tuned such that the mesopause height at *z* ≈ 80 km agrees with HWM2014. In general, MS-GWaM-1D and MS-GWaM-3D produce physical zonal-mean zonal winds with the expected wind reversals in both the summer and winter hemispheres (Figs. 2b–d,f–h). As one may expect, runs with the same model setup but without any nonorographic parameterization are subject to strong jets and entirely lack the mesopause reversal (Figs. 2a,e).

Notable differences to the HWM2014 can be seen in the summer hemispheric jets, in the tropical stratosphere, and in the Antarctic winter jet. Comparing the two MS-GWaM results with the run without any nonorographic gravity wave drag parameterization (Figs. 2a,e), we find that the tropical stratospheric jet at *z* ≈ 50 km height is a robust feature of the current model version, which is present in all three runs. The aforementioned tuning runs indicated that there is little to no influence on the structure by the tuning parameters (not shown). We thus conclude that it is generally independent of the gravity wave parameterization. The weak northern summer jet, however, seems to be related to the transient parameterization in general, as it is present in runs with both MS-GWaM-1D and MS-GWaM-3D (Figs. 2b,c). While we note a similar bias in the work of Bölöni et al. (2021), the runs without any nonorographic gravity wave parameterization do not suffer from this bias (Fig. 2d). Finally, we observe that the slant of the Antarctic winter jet toward the equator is not entirely reproduced by MS-GWaM-3D (Fig. 2c). This is due to a modified zonal and meridional gravity wave drags in the Southern Hemisphere above 60-km altitude in MS-GWaM-3D as opposed to MS-GWaM-1D as will be discussed in section 4c. Possible reasons for the modified wave drag might be improper spectral characteristics of the current background source, errors near the pole due to the representation of the ray-volume extent in spherical coordinate systems (cf. section 3a), or even codependencies of the columnar gravity wave parameterization with other parameterizations such as the employed orographic wave drag parameterization. Moreover, other components of the model, such as the radiation scheme or the modified ozone climatology, are tuned to produce realistic temperature distributions in combination with the standard orographic wave drag. In general, the confidence in the present implementation is high, as MS-GWaM-1D was validated by Bölöni et al. (2021) and Kim et al. (2021) and MS-GWaM-3D was validated against the results of Wei et al. (2019) using an idealized cubic geometry implemented in ICON (not shown). However, a more detailed analysis may be needed to identify the reasons behind the phenomenon and mend the setup accordingly.

Similar to the zonally averaged zonal winds, we find that the zonally averaged temperatures and the corresponding MSIS fields agree well (Fig. 3). In particular, the MS-GWaM reproduces the cold summer pole at the mesopause and the warm winter pole associated with the stratopause (Figs. 3b,c,f,g). These features are naturally not present when the nonorographic gravity wave drag parameterization is switched off (Figs. 3a,e).

In general, the zonal mean model state generally seems to be moderately affected by enabling three-dimensional wave propagation in MS-GWaM. While the runs with the transient parameterizations generally exert some bias in the northern summer hemisphere, the Antarctic winter jet is too strong at altitudes around 50–80 km for MS-GWaM-3D. Runs for December do not contain these systematic biases. This resembles the findings of Bölöni et al. (2021) where a weak dependency of the zonal mean wind on the transience of the gravity wave parameterization was detected. Albeit somewhat counterintuitive, it may not be a surprise, as the spatial and temporal averages represent the quasi-steady state of the model setups. These, in turn, are tuned to approximate a similar zonal mean model state. It was, however, also shown that both the transience and the localization of the gravity waves can have significant impacts on the structure of the horizontal mean flow (e.g., Sato et al. 2009; Šácha et al. 2016; Ehard et al. 2017). For instance, the mean-state results may change if a more realistic spatiotemporal distribution of the wave source spectrum was employed. While the investigation of realistic source distribution is left for future work, in the following sections, we focus on the nature of three-dimensional propagation of parameterized IGWs, examining wave action fluxes and budgets and horizontal structures of IGW momentum fluxes and mean wave drags.

### b. Global three-dimensional IGW distribution

Gravity waves are generally three-dimensional phenomena that are present in large parts of the atmosphere. While wave-resolving simulations can provide insights into global wave activity, they typically span short model times due to the limitations of both computational resources and required storage. Additionally, the separation of IGWs from other dynamics and their characterization are generally nontrivial tasks. Being based on a multiscale WKBJ theory (Achatz et al. 2017), MS-GWaM simulates the global behavior of nonorographic gravity waves without the need to explicitly resolve their fast varying phases. As a result, it can be used to predict global distributions of phase-space wave action densities and derived quantities such as momentum fluxes and the resulting wave drag. As an example, we analyze the global wave action budget and augment it with the zonal wave drag to illustrate the resulting effect on the mean flow.

*t*. Here, the three right-hand side terms correspond to the temporally averaged wave action tendency contributions from the horizontal wave propagation, the vertical wave propagation, and the wave saturation. Moreover, the left-hand term is small for sufficiently long averaging intervals. For a sufficiently small left-hand side of Eq. (30) while preserving the planetary wave structure, we choose Δ

*t*= 10 days and show the predictions of the right-hand side contributions by MS-GWaM-1D and MS-GWaM-3D at altitudes of approximately 25 and 60 km for 10–20 June 1991. While the 25-km level shows the stark differences between the 1D and 3D schemes, the wave dynamics at 60-km altitude highlight the interaction with the Antarctic polar night jet in more detail.

As one may expect, MS-GWaM-1D predicts a balance between vertical wave action propagation and dissipation at either altitude (Figs. 4c,e and 5c,e). Thus, wave breaking and strong mean flow impacts can only occur where IGWs were previously able to vertically propagate through the wind shear of the underlying air column. Negative values for the change in wave action due to vertical propagation are associated with a finite temporal averaging interval Δ*t*. For instance, propagating mean flow structures such as planetary waves may lead to changes in the wave action budget, which adjust only on the time scale of the wave propagation. Although not linearly related, the strongest zonal wave drags occur where the wave dissipation is high. At a rather low altitude of 25 km, this results in a rather uniform distribution of wave dissipation and drag in the Southern Hemisphere, accompanied with intermittent convective events in the Northern Hemisphere (Figs. 4e–g). Running the same case with MS-GWaM-3D, we find that both the contributions due to the horizontal and vertical propagation are of similar magnitude at an altitude of 25 km, such that the resulting balance includes all terms (Figs. 4b,d,f). Near horizontally sheared wind structures, such as the edges of the Antarctic winter jet, the two contributions from the horizontal and vertical propagation form inverted dipoles associated with wave refraction phenomena. In the case of the mentioned Antarctic winter jet, as we will show below, this corresponds to the often observed wave refraction into the jet (e.g., Ehard et al. 2017; Hindley et al. 2020). Interestingly, the wave refraction follows the structure of the resolved planetary waves and ultimately leads to a shift of lower-level wave action dissipation toward regions of strong horizontal shear, as, for instance, found between the summer and winter jets (Fig. 4f). Despite the different launch amplitudes (see section 3c), all contributions have similar magnitudes on both hemispheres. Correspondingly, the horizontal propagation significantly modifies the zonal wave drag at 25 km. While the Southern Hemisphere drag is much reduced, the Northern Hemisphere exhibits comparatively strong drags near planetary wave structures. Interestingly, the wave action budgets change structurally with altitude. At *z* ≈ 60 km, both the budgets of MS-GWaM-1D and MS-GWaM-3D are mostly controlled by vertical wave action propagation and dissipation (Figs. 5c–f). In contrast to the lower altitude, horizontal wave action propagation is much reduced. However, horizontal propagation through lower altitudes shifts the action dissipation and wave drag near the southern winter jet toward the equator (Figs. 5e–h). We suspect this shift to cause the missing wave drag at higher altitudes near the pole, leading to a biased jet structure in the zonal mean. These results suggest an important influence of the horizontal propagation near the Antarctic winter jet, and thus, we analyze the zonal mean drag structures in the following.

### c. Wave propagation and drag in the vicinity of horizontally sheared jet structures

Expanding on the horizontal refraction behavior found in the horizontal maps (Figs. 4 and 5), we show zonal averages of wave action fluxes (Fig. 6) and the corresponding wave drag (Fig. 7) temporally averaged over free runs for June and December 1991 through 1998. By construction, MS-GWaM-1D only allows for vertical propagation (Figs. 6a,c). The corresponding wave action tendency is thus associated with the vertical wave action flux convergence. The consequently generated wave drag accelerates the zonal wind structures, with maxima occurring at the mesopause (Figs. 7a,c). Note that the maximum wave action flux convergences are not equivalent to the maximum wave drag due to the scaling with the wavevector (cf. section 2). MS-GWaM-3D shows strong horizontal propagation in regions of strong horizontal and vertical shear, such as jet edges (Figs. 6b,d). Notably, in the Southern Hemisphere winter, the wave action flux vector points northward near the South Pole, i.e., into the southern winter jet (Fig. 6b). Consequently, the polar wave action fluxes are reduced at altitudes near the mesopause and the wave drag maximum is weaker south of 75°S (Figs. 6b and 7b). Moreover, the equatorial regions are strongly affected by the horizontal wave propagation (Figs. 7b,d). Here, the wind shear is generally strong, but the mean wave action fluxes are weak due to the intermittency of the convective wave source.

To highlight the dynamics near the Antarctic winter jet, we additionally show the zonal and meridional wave drags for both MS-GWaM-1D and MS-GWaM-3D near the jet, averaged for all June (Fig. 8). Here, the reduced zonal wave drag for MS-GWaM-3D, south of 75°S, becomes particularly visible (Figs. 8a,b). Moreover, the maximum zonal wave drag is shifted southward due to the horizontal propagation. Stronger differences can be seen for the generally weaker meridional component (Figs. 8c,d). In particular, it changes sign and exerts a southward drag on the jet when enabling horizontal propagation. Note that this behavior is only observed above the jet. Globally, the meridional drag accelerates the flow from the winter toward the summer hemisphere for both models (not shown). It remains unclear, however, whether the modeled meridional drag is a consequence of the too strong winter jet over the pole or vice versa. For instance, the ray-volume discretization, the interaction with the orographic wave drag parameterization, or the structure of the modeled temperatures may be a cause of the biased wind structures (also compare section 4a and Fig. 2c). Further and more targeted investigation may be needed to bring clarity about this effect.

Differences in refraction behavior can also be seen in the summer hemispheres. However, the model biases of the northern summer hemispheric jet structures (cf. section 4a) complicate the interpretation of the wave action budgets in the corresponding regions.

One may now ask the question how realistic the modeled meridional wave propagation near the Antarctic winter jet is and how it corresponds to satellite observations. To provide some insight, we show the meridional momentum fluxes [cf. Eq. (12)] from the composites of June for the years 1991 through 1998 at altitudes ≈ 40 km (Fig. 9). Note that the meridional fluxes of MS-GWaM-1D must be interpreted keeping in mind the assumed local horizontal homogeneity. That is, albeit the wave parameters suggest horizontal momentum fluxes, the horizontal propagation (including the wave advection) is set to zero in the columnar setup. The resulting *synthetic* meridional fluxes seem to be unrelated to the underlying zonal jet structure (Fig. 9a). As waves are launched homogeneously in all directions at the lower boundary (section 3c), the resulting meridional fluxes are a product of the partial wave filtering due to local vertical gradients of both the horizontal wind and the buoyancy frequency. In contrast, MS-GWaM-3D shows meridional convergence of wave momentum fluxes into the polar jet (Fig. 9b). In particular, the meridional fluxes are northward over the Antarctic continent and southward further north. The change in sign, i.e., the convergence zone, roughly coincides with the zonal wind maximum. This convergence effect of internal gravity waves into the polar night jets is well documented from both observations and general circulation models (e.g., Dunkerton and Butchart 1984; Sato et al. 2012; Ehard et al. 2017; Hindley et al. 2020; Gupta et al. 2021). Notably, the structure and amplitudes of the meridional momentum fluxes from MS-GWaM-3D simulations resemble the observations of Hindley et al. (2020; compare their Fig. 3d). Naturally, MS-GWaM cannot reproduce the strong momentum fluxes originating from orographic waves over the Antarctic Peninsula, the Southern Andes, or the islands of the Southern Ocean. The otherwise fairly close fit, however, gives us high confidence in the correctness of MS-GWaM-3D.

## 5. Discussion of achievements and challenges

Although IGWs are an essential part of atmospheric dynamics, their parameterization in general circulation models is typically subject to various simplifications. In particular, the single-column assumption, the steady-state approximation, and the balanced mean flow assumption have usually been made. Here, we present a MS-GWaM-3D, which aims at parameterizing IGWs through Lagrangian ray tracing without the three mentioned assumptions. In general, we find that including 3D transience significantly impacts global IGW propagation patterns and thus horizontal IGW distributions. Correspondingly, the associated wave drag and mean flows are modified.

We compare two distinct situations: For MS-GWaM-1D, where the columnar approximation is applied, the vertical propagation is balanced by wave breaking in the wave action equation (Figs. 4c,e and 5c,e). Only waves that did not encounter critical layers according to their wave properties may propagate to higher altitudes and eventually break. In contrast, MS-GWaM-3D allows for modulated and spatially unconstrained propagation and enables waves to be refracted around wind structures. As a consequence, both horizontal and vertical propagation balance with the wave dissipation (Figs. 4b,d,f and 5b,d,f). The equal order of magnitude in the contributions of horizontal and vertical wave propagation to the wave action budget emphasizes the importance of including horizontal propagation. Moreover, it questions the validity of using columnar methods when, e.g., studying IGW distributions. Both methods do, however, perform as expected in reproducing the cold summer pole and the warm winter pole at altitudes ∼85 and ∼60 km, respectively, in the climatological zonal mean (Fig. 3). The corresponding wind reversals and the mesopause altitudes are reasonably predicted (Fig. 2). We are thus confident that MS-GWaM-3D, as MS-GWaM-1D before it, covers some major effects of IGWs on the mean flow dynamics, rendering MS-GWaM-3D a viable IGW parameterization. The 3D wave propagation is also found to have significant impacts on the variability of the zonal mean flow, such as the quasi-biennial oscillation (QBO), as presented by Kim et al. (2023).

Including the horizontal propagation does, however, also introduce some important differences in the simulated climatology. In particular, the Southern Hemisphere winter jet becomes stronger near the pole at altitudes 60–80 km for MS-GWaM-3D (Fig. 2c). The reason behind this change is, however, not obvious. One possible explanation is the northward refraction of gravity waves near the Antarctic winter jet in MS-GWaM-3D as compared to MS-GWaM-1D. Both the 3D wave action tendencies (dipole structures in Figs. 4d,f) and the zonally averaged wave action fluxes (vectors in Figs. 6a,b) suggest that the 3D modulation causes IGWs to propagate northward and thus relate to a weaker gravity wave drag at high altitudes and similar latitudes (Figs. 8a,b). Additionally, the meridional wave drag near the jet is oppositely directed, that is southward, for MS-GWaM-3D (Figs. 8c,d). It remains unclear, however, whether the wave refraction and ultimately the wave drag cause the shifts of the jets as the two structures a nonlinearly coupled. Possible other causes are the numerical simplifications made in the setup of the ray tracer in the spherical coordinate system or codependencies of the gravity wave parameterization with other parameterizations, for instance, the orographic gravity wave drag. Finally, the jets are sensitive to the structure of the modeled temperature distribution, which, in turn, is dependent on, for instance, the ozone climatology and the radiation scheme. A thorough investigation will be needed, and we hope to answer these questions in an envisioned follow-up study. Finally, we find that the meridional wave propagation in MS-GWaM-3D around the Antarctic polar night jet resembles the observations of Hindley et al. (2020) both in amplitudes and in structure, up to orographic wave sources such as the southern Andes (Fig. 9). This gives us high confidence into the implementations as the convergence into the polar night jet is well documented in a number of studies (e.g., Dunkerton and Butchart 1984; Sato et al. 2012; Ehard et al. 2017; Hindley et al. 2020; Gupta et al. 2021).

There is a long list of effects that may be analyzed in more detail from this starting point. Notably, the role of IGWs in sudden stratospheric warmings (e.g., Baldwin et al. 2021), the accuracy of the final warming date (e.g., De la Cámara et al. 2016; Eichinger et al. 2023), the missing wave drag at 60°S (e.g., Holt et al. 2023), etc., are of interest and are planned for future simulations and analyses. Current investigations include the impact of 3D transient waves on the quasi-biennial oscillation and wave intermittency. Moreover, there are efforts into increasing the realism of the included IGW sources related to jet frontal systems and flow over topography. Finally, code optimization may reduce the computational cost, which amounts to a runtime factor ∼30 with respect to runs without any nonorographic wave drag parameterization (cf. section c in appendix B). With these challenges in mind, we ultimately aim for the application of MS-GWaM-3D in climate simulations and possibly numerical weather predictions.

Furthermore, there are numerous potential improvements that may be added to MS-GWaM in the future, albeit not considered here. Most importantly, MS-GWaM-3D—for the time being—does not include a flow-dependent description of the emission of IGWs from jets and fronts (Charron and Manzini 2002; Richter et al. 2010; De la Cámara and Lott 2015) or flow over topography (Palmer et al. 1986; Bacmeister et al. 1994; Lott and Miller 1997; Xie et al. 2020, 2021; van Niekerk and Vosper 2021; van Niekerk et al. 2023; Eichinger et al. 2023). For the latter reason, MS-GWaM needs augmentation with a subgrid-scale orography parameterization. Moreover, it is not clear how important modulated triadic resonant interactions among IGWs or between subgrid-scale geostrophic modes and IGWs are for atmospheric dynamics, which could, however, be investigated through ray-tracing techniques (cf. Kafiabad et al. 2019; Voelker et al. 2021).

Revisiting past studies, we recognize that Ribstein et al. (2015) and Ribstein and Achatz (2016) omit some of the important metric terms so that their results should be interpreted with care.

## Acknowledgments.

UA thanks the German Research Foundation (DFG) for partial support through the research unit “Multiscale Dynamics of Gravity Waves” (MS-GWaves; Grants AC 71/8-2, AC 71/9-2, and AC 71/12-2) and CRC 301 “TPChange” (Project 428312742 and Projects B06 “Impact of small-scale dynamics on UTLS transport and mixing” and B07 “Impact of cirrus clouds on tropopause structure”). YHK and UA thank the German Federal Ministry of Education and Research (BMBF) for partial support through the program Role of the Middle Atmosphere in Climate (ROMIC II: QUBICC) and Grant 01LG1905B. UA and GSV thank the German Research Foundation (DFG) for partial support through the CRC 181 “Energy transfers in Atmosphere and Ocean” (Project 274762653 and Projects W01 “Gravity-wave parameterization for the atmosphere” and S02 “Improved Parameterizations and Numerics in Climate Models”). UA is furthermore grateful for the support by Eric and Wendy Schmidt through the Schmidt Futures VESRI “DataWave” project. This work used resources of the Deutsches Klimarechenzentrum (DKRZ) granted by its Scientific Steering Committee (WLA) under Project bb1097.

## Data availability statement.

All data used in this publication may be available upon request to the corresponding author.

## APPENDIX A

### The Gravity Wave Forcing of the Mean Flow and Its Relation to the EP Flux Formulation

As laid out in section 2, MS-GWaM-3D utilizes the perturbation flux formulation as introduced by Wei et al. (2019). The interested reader may note that the resulting mean flow impact differs from the commonly used pseudomomentum flux convergence related to the Eliassen–Palm flux (Eliassen and Palm 1961; Andrews and McIntyre 1976). To highlight the relation to the latter, we would first like to remind the reader about the underlying assumption of the multiscale theory applied here. First, the theory is built on the compressible Euler equations such that nonlinearities not only arise from the advection operators but also from the pressure gradient terms. As MS-GWaM acts as a gravity wave parameterization for the underlying Eulerian model, phase averages are constructed in a Eulerian sense, accordingly. In terms of scale separation, it is assumed that horizontal scales are much larger than vertical scales. Furthermore, the Brunt–Väisälä frequency is assumed to be much larger than the Coriolis frequency. These assumptions then lead to the regime of small Rossby numbers with a leading-order mean flow in hydrostatic and geostrophic balance. Finally, all wave scales are assumed to be much smaller than the synoptic mean flow scales.

*ψ*. Note that the latter relates to the leading-order synoptic-scale wind through

**U**=

**e**

*× ∇*

_{r}*and that within geostrophic theory also the thermodynamic fields can be obtained to leading order from the streamfunction. Hence, as long as the mean flow is within the synoptic scaling regime, it is controlled completely by Eq. (A6). Furthermore, we have denoted the horizontal pseudomomentum fluxes*

_{r}ψSince the zonal means of zonal wind and temperature are to a good approximation in geostrophic and hydrostatic balance, we therefore do not find a statistically significant difference in those fields. However, Wei et al. (2019) showed that the local transient response of the Eulerian phase averaged wind in the presence of a finite amplitude gravity wave packet becomes more accurate when considering the fluxes as denoted in Eqs. (A1) and (A2). We thus implement this formulation but leave the in-depth analysis of the local wind response to an envisioned follow-up study.

## APPENDIX B

### Details of the Numerical Implementation

To simulate transient internal gravity waves, MS-GWaM uses a spectral extension of WKBJ theory (Bretherton 1966; Grimshaw 1975; Achatz et al. 2017; Achatz 2022; and references therein). A Lagrangian approach is employed to solve the equations numerically (Muraschko et al. 2015; Bölöni et al. 2016, 2021; Kim et al. 2021). Therein, the six-dimensional phase space, spanned by position and wavenumber, is split into small finite-size ray volumes, propagating according to the eikonal equations [Eqs. (3)–(6)]. They keep their regular shape of rectangles in (*λ*, *ϕ*, *r*, *k _{λ}*,

*k*,

_{ϕ}*k*), but their extent in all these six directions is allowed to vary in a volume-preserving manner. The wave action density carried by each ray volume is constant, unless wave breaking leads to the onset of turbulence and hence a decrease in

_{r}#### a. Three-dimensional field interpolation

*N*, and the mean winds

**U**are the functions of space and differ between the four faces forming with the meridional and zonal boundaries of the ray volume (see crosses in Fig. B1). To account for a corresponding change in the area of the ray volume, we interpolate all needed background fields horizontally to the face center positions and the ray-volume center through linear Taylor expansions as follows:

*χ*symbolizes the interpolated field and the subscripts cc and

*j*refer to evaluations at the cell center and the ray-volume face or ray-volume center, respectively. In the vertical, we apply linear interpolations between the grid points. While the tendencies for the wavenumbers are computed with the ray-volume centered values, the interpolated fields at the ray-volume faces are used to compute group velocities, enabling the prediction of the changing extent of the ray volume in physical space [Eqs. (1) and (2)]. In particular, the difference in the group velocities at the center of opposing ray-volume faces predicts the rate of compression along the corresponding direction. The average of the opposing group velocities is then used to predict the change in location along the characteristic [Eq. (3)]. Consequently, the change in the wavenumber extents Δ(

*k*,

_{λ}*k*,

_{ϕ}*k*) is determined following the procedure described in section 3a.

_{r}#### b. Horizontal propagation and code parallelization

Since MS-GWaM propagates Lagrangian ray volumes through space, it cannot easily be parallelized using the traditional MPI infrastructure of the ICON model developed for the synchronization of Eulerian fields. In particular, the ray volumes may propagate over large distances and transfer between parallel MPI domains. To accompany this while avoiding load balancing problems, we associate each ray volume with a parent cell based on its current location and store it in a corresponding grid-based array. As an example, in the shown sketch (Fig. B1), the hatched ray volume is associated with the white-bounded underlying cell. It may, however, have a group velocity such that it propagates into a new parent cell, represented by the arrow and the dark gray shaded triangle, respectively. Therefore, after each propagation step, all cells such as the dark gray shaded triangle search all rays in all neighboring cells (light gray shading) and transfer those that are now located in their own cell area. With this construction, we can localize arrays of ray volumes based on the Eulerian grid column. The integration along the wave characteristics, launch processes, saturation calculations, and wave projection onto the Eulerian grid can then be parallelized using the domain decomposition of the dynamical core with the given model infrastructure. Note that the modulation equations are integrated using a three-step Runge–Kutta low-storage scheme (Williamson 1980).

#### c. Ray-volume splitting

*j*vertically, which become larger than the local thresholds:

*h*and a scale factor

_{c}*α*(which is set to 2.5 in this study). The volume is cut into two equally sized ray volumes in the vertical, with the wavevectors identical to the original parent wave. The carrier ray location of each split volume is then set to its center. Similarly, ray volumes are split horizontally where either of the three thresholds is reached:

_{s}*a*represents the local cell area and

_{c}*R*represents Earth’s mean radius. Splitting is done along the direction of the larger extent.

#### d. Horizontal flux smoothing

To avoid numerical artifacts based on the employed simplification in the ray volume representation, we smooth all diagnostic outputs and wave fluxes. Each triangular cell combines three vertices, one in each corner (cf. Fig. B1). Each of these vertices is connected to five or six neighboring cells, depending on the location of the triangulated icosahedron. The smoothing algorithm first averages the corresponding fields over these pentagons or hexagons and then combines the three resulting values into one average. Consequently, the weights are distributed such that the center cell is weighted with a factor 3, the direct adjacent cells with one common edge are weighted with a factor 2, and all neighboring cells sharing a vertex with the center cell are weighted with a factor 1. While this procedure generally reduces the amplitude of strong flux peaks, it also reduces the occurrence of grid-scale noise and ensures model stability.

#### e. Computational performance

Finally, conceptional models need to prove themselves not only through the accurate representation of not-resolved physics but also through competitive performance. In particular, ray-tracing models may suffer from load balancing problems due to the uneven distribution of ray integrations between processors but also the existence of too many rays in general. As described in section b of appendix B, MS-GWaM utilizes a cell-based representation of ray volumes, which is highly parallelizable such that load balancing problems may not inhibit the model performance. Additionally, the total number of ray volumes is restricted by the maximum numbers *n*_{bg} ≤ 4800 and *n*_{conv} ≤ 8800 per model-grid column for background and convective gravity waves, respectively. For the lowest impact, the ray volumes with the lowest wave energy are chosen to be removed. Detailed analyses show that the impact of this procedure is indeed small and may be neglected (not shown).

To measure the performance of the model, we compare the runtimes of all 2-month runs without any nonorographic gravity wave parameterization with both the MS-GWaM-1D and MS-GWaM-3D setups (Fig. B2). Additionally, we show the fraction of runtime that is associated with MS-GWaM and normalize all values with the number of simulated days. These measures show that MS-GWaM-1D and MS-GWaM-3D slow down ICON by factors of approximately 12 and 30, respectively. That is, the 3D setup has a performance penalty equal to a factor of approximately 2.5. Still, the gravity wave parameterization is several orders of magnitude faster than wave-resolving simulations—for which Bölöni et al. (2021) estimate a cost factor of about six orders of magnitude—and thus a competitive tool for the estimate and analysis of global IGW effects.

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