MS-GWaM: Multi-Scale Gravity-Wave Model

• MS-GWaM is a multi-scale gravity-wave model that uses a Lagrangian, ray-based phase-space description to simulate subgrid dynamics with finite propagation speed.
• It captures transient gravity-wave propagation, incorporating horizontal and vertical wave refraction and direct coupling with mean atmospheric flows.
• Its numerical implementation via discretized ray volumes in ICON and orographic applications reduces wind profile errors and improves momentum and energy flux computations.

The Multi-Scale Gravity-Wave Model (MS-GWaM) is a prognostic gravity-wave parameterization and wave–mean-flow framework for atmospheric models that replaces the classical combination of single-column and steady-state assumptions with a Lagrangian, ray-based, phase-space description of subgrid-scale gravity waves. In MS-GWaM, gravity-wave packets are propagated along rays with finite travel time, their spectral wave action is transported in physical and wavenumber space, and their momentum, entropy, and pseudomomentum effects can be coupled back to the resolved circulation. The framework was developed from nonlinear WKB theory, extended from transient columnar formulations to three-dimensional propagation, and subsequently applied to idealized orographic waves, global non-orographic waves, wave–turbulence coupling, and the interaction of gravity waves with middle-atmosphere mean flow and solar tides (Achatz et al., 2023, Voelker et al., 2023, Jochum et al., 2024).

1. Asymptotic and dynamical foundations

MS-GWaM is rooted in a multi-scale asymptotic theory for the interaction between small-scale gravity waves and a larger-scale balanced flow. In the theoretical formulation, the atmosphere is treated as compressible, rotating, and hydrostatic on an $f$-plane, with a hydrostatic reference state $\bar\theta(z)$, $\bar\rho(z)$, and $\bar\pi(z)$. Scale separation is introduced through a small parameter $\varepsilon \sim O(10^{-1})$, with synoptic-scale flow varying slowly relative to the gravity waves. The theory explicitly distinguishes weak stratification $(\alpha=1)$ from moderately strong stratification $(\alpha=0)$, which allows the asymptotics to remain valid in the middle atmosphere rather than being restricted to classic quasi-geostrophic limits (Achatz et al., 2023).

The wave field is represented by a WKB ansatz in which the rapidly varying phase $\phi(X,T)$ defines the local wavenumber and frequency,

$k=\nabla_X \phi, \qquad \omega=-\partial_T \phi.$

At leading order, the resolved mean flow is horizontal, hydrostatic, and geostrophic, while the wave field obeys the intrinsic hydrostatic-inertia-gravity-wave dispersion relation

$$\Omega(k,X,T)=k\cdot U_0^{(0)} \pm \sqrt{\frac{N_0^2 k_h^2 + f_0^2 m^2}{k_h^2+m^2}}.$$

This formulation is not merely a kinematic background for rays. The balanced synoptic-scale flow is itself a dynamical component that exchanges momentum, heat, and potential vorticity with the wave field (Achatz et al., 2023).

A key distinction in the theory is between a fully nonlinear regime at strong wave amplitudes and a quasilinear spectral regime for weak gravity-wave amplitudes. The latter is the practical basis for MS-GWaM as a parameterization: multiple spectral components can be superposed, each retaining its own transport and refraction dynamics. This spectral generalization is what makes the framework suitable for atmospheric-model implementations that must represent ensembles of unresolved waves rather than a single monochromatic disturbance (Achatz et al., 2023).

2. Phase-space transport, conserved quantities, and mean-flow coupling

In MS-GWaM, the fundamental prognostic quantity is the phase-space wave-action density, written as either $\bar\theta(z)$0 or $\bar\theta(z)$1 depending on the formulation. The wave field is advanced in phase space rather than in a single vertical column. Rays evolve according to the eikonal equations

$\bar\theta(z)$2

with group velocity

$\bar\theta(z)$3

For the spectral density, the core transport law is

$\bar\theta(z)$4

or, because the phase-space velocity is non-divergent, equivalently

$\bar\theta(z)$5

This is the central mathematical structure of MS-GWaM: a conservative advection problem in phase space that retains finite propagation speed, horizontal propagation, refraction, and time dependence (Achatz et al., 2023, Jochum et al., 2024).

Wave-action conservation is inherited from the underlying WKB theory. In the locally monochromatic setting,

$\bar\theta(z)$6

The theory also includes explicit conservation statements for energy and an extended potential-vorticity-like invariant,

$\bar\theta(z)$7

which is conserved in the absence of sources, sinks, and dissipation. This extended PV conservation clarifies when gravity waves can, and cannot, accelerate the mean flow. In particular, the framework recovers the logic behind the non-acceleration theorem while showing that many classical parameterizations enforce it only after imposing stronger assumptions such as stationarity, columnarity, and the neglect of thermodynamic forcing (Achatz et al., 2023).

The coupling to the resolved flow can be expressed through momentum fluxes, entropy fluxes, or pseudomomentum fluxes, depending on the configuration. A central point in later MS-GWaM applications is that conventional Eliassen–Palm-type forcing is formally appropriate only when the resolved flow is in geostrophic and hydrostatic balance. MS-GWaM can instead use direct momentum-flux and entropy-flux forcing, which is particularly relevant for the mesosphere and lower thermosphere, where solar tides are strong and ageostrophic. This generalization is one of the reasons the framework has been used to examine “non-classical” gravity-wave dynamics in ICON (Kühner et al., 26 Sep 2025).

3. Numerical realization and major model variants

The practical implementation of MS-GWaM follows directly from its phase-space structure. The spectral wave-action equation can be solved Eulerianly, but the papers emphasize a more efficient Lagrangian realization in which phase space is discretized into finite ray volumes. Each ray volume is advected in $\bar\theta(z)$8-space according to

$\bar\theta(z)$9

and, because phase-space volume is conserved, the ray volume is deformed but not created or destroyed. In global implementations, fluxes are evaluated at ray-volume centers and then projected onto the faces of the resolved-flow finite volumes (Achatz et al., 2023, Voelker et al., 2023).

Wave dissipation and breaking are represented by saturation schemes tied to static instability. In the theoretical review, breaking is invoked when the combined perturbation drives the total vertical potential-temperature gradient negative, and a dissipative sink is applied to the wave action (Achatz et al., 2023). In the idealized orographic application, the sink is written as

$\bar\rho(z)$0

which acts as a turbulent-diffusion representation of saturation and breaking (Jochum et al., 2024).

Several named configurations recur across the literature:

Variant	Core assumption	Reported use
MS-GWaM-1D	Transient but columnar	Earlier ICON implementation for internal gravity waves (Voelker et al., 2023)
MS-GWaM-3D	Full 3D transient propagation	Global non-orographic waves in ICON (Voelker et al., 2023)
WKB-TR	Fully transient WKB/MS-GWaM	Idealized orographic tests in PincFlow (Jochum et al., 2024)
WKB-ST	Steady-state WKB/MS-GWaM	Classic mountain-wave comparator (Jochum et al., 2024)

The global 3D ICON implementation includes a latitude-dependent background source and convectively generated waves. The background source is launched at 300 hPa and divided among four horizontal directions. The convective source is coupled to the convection parameterization; in the simulations described in the 3D extension paper, the areal fraction of latent heat release is set to $\bar\rho(z)$1 (Voelker et al., 2023). By contrast, the idealized orographic experiments are conducted in PincFlow, with two-dimensional, isothermal flow over a monochromatic lower boundary

$\bar\rho(z)$2

mountain heights from $\bar\rho(z)$3 m to $\bar\rho(z)$4 m, a rigid upper boundary, a Rayleigh damping layer aloft, and a linear three-hour spin-up of the orography (Jochum et al., 2024).

4. Relation to classical gravity-wave parameterizations

MS-GWaM was developed explicitly against three approximations that dominate conventional gravity-wave parameterizations: the single-column approximation, the steady-state assumption, and, in many formulations, the reduction of wave forcing to pseudomomentum or Eliassen–Palm flux convergence. Traditional schemes therefore tend to force only the momentum equation, ignore wave thermodynamic forcing, neglect horizontal propagation, and treat the wave field as if it instantaneously adjusted to the current background state (Achatz et al., 2023, Kühner et al., 26 Sep 2025).

The difference is especially clear in the orographic application. In the steady-state implementation, all time derivatives are suppressed and the wave field is treated as if it were instantaneously distributed in the vertical. The lower-boundary source sets the horizontal wavenumbers, stationarity diagnoses the intrinsic frequency through

$\bar\rho(z)$5

and wave action is simply set to zero above levels where $\bar\rho(z)$6, i.e. at diagnosed critical levels. In the transient implementation, by contrast, waves are launched, propagated, refracted, and dissipated with finite travel time, so the resolved flow responds only after the wave packets have actually reached a given altitude (Jochum et al., 2024).

A recurring misconception in older parameterization logic is that horizontal propagation is a secondary correction to vertical propagation. The global 3D MS-GWaM study argues otherwise: in the wave-action budget, horizontal wave propagation is as important as vertical wave propagation, and the corresponding refraction includes effects such as wave steering into polar jet streams. This modifies where dissipation occurs, where drag is deposited, and how zonal-mean winds are shaped (Voelker et al., 2023).

Another misconception is that critical levels necessarily truncate wave forcing in the same way in transient and steady descriptions. The idealized mountain-wave study states that the fully transient theory has no persistent critical levels in the same sense as the steady-state reduction, which is why the transient model can continue to force the flow above levels where the classic model shuts off the wave. In that setting, the choice between transient and steady formulations determines not only the magnitude of drag but the qualitative possibility of high-altitude wind reversal (Jochum et al., 2024).

5. Orographic-wave application and the role of transience

The first application of MS-GWaM to orographic gravity waves used idealized two-dimensional flows over monochromatic mountains in PincFlow and compared three configurations: WRS wave-resolving simulations without MS-GWaM, WKB-TR low-resolution simulations with the transient model, and WKB-ST low-resolution simulations with the steady-state model (Jochum et al., 2024).

For low mountains, the key result is that the steady-state approximation spreads wave influence through the entire domain almost immediately. This produces unrealistic early deceleration and can even generate static instability high above the surface. The transient model, by contrast, reproduces the slow upward propagation seen in the wave-resolving simulations: low-level flow is affected first near the source, and only later does the forcing appear aloft as the wave packet ascends. In momentum-flux Hovmöller diagnostics and mean-wind profiles, the transient solution follows the resolved simulation much more closely, and the root-mean-square error of the mean wind is reported to be consistently smaller, sometimes by more than a factor of two in the linear regime (Jochum et al., 2024).

For high mountains, the difference becomes dynamically stronger. The wave becomes nonlinear near the lower boundary, breaks at high altitude, and in the wave-resolving simulation the resulting forcing can drive a wind reversal aloft. The transient MS-GWaM configuration captures this reversal, whereas the steady-state configuration cannot, because its critical-level assumption removes wave action once the intrinsic frequency tends to zero. The paper identifies this as one of the strongest arguments for a fully transient orographic parameterization: the timing of propagation relative to saturation and breaking determines whether the mean flow experiences ordinary drag or a more dramatic reversal of the background wind (Jochum et al., 2024).

The physical interpretation given in the study is that mountain waves do not instantaneously fill the domain. Instead, they are launched at the surface, propagate slowly upward, undergo gradual refraction, eventually break, and only then can reverse the upper-level wind. The paper therefore frames orographic gravity-wave forcing not as a function of height alone, but as a strongly time-dependent process controlled by finite propagation speed and evolving wave–mean-flow interaction (Jochum et al., 2024).

6. Global applications, middle-atmosphere impacts, and related directions

In its global 3D extension, MS-GWaM was implemented in ICON as a parameterization for subgrid-scale non-orographic internal gravity waves. The study reports that the model contains three-dimensional transient propagation with accurate flux calculations, a latitude-dependent background source, and convectively generated waves, and that it reproduces expected temperature and wind patterns in the mesopause region in the climatological zonal-mean state. A global wave-action-budget analysis shows that horizontal wave propagation is as important as vertical wave propagation, and that three-dimensional refraction redistributes wave action into structures such as the polar jet streams, thereby modifying zonal-mean wave drag and zonal-mean winds (Voelker et al., 2023).

A later UA-ICON study used MS-GWaM to quantify two processes that standard schemes typically omit: oblique propagation and explicit bidirectional coupling between gravity waves and turbulence. Allowing oblique propagation alone lowers and cools the summer mesopause by shifting momentum and heat deposition to lower altitudes; the strongest residual-mean upwelling and equatorward motion shift downward by about 10 km, the summer mesopause drops to around 85 km over the South Pole in the shown cases, gravity-wave-induced vertical shear is reduced by at least 90% in places, and turbulent kinetic energy in the upper mesosphere and lower thermosphere can increase by nearly 70%. When explicit wave–turbulence coupling is added, the response becomes nearly opposite in the mesopause region: the summer mesopause rises to about 90 km, warms from about 107 K to about 112 K, gravity-wave-induced shear remains reduced over much of the atmosphere, TKE near the summer mesopause rises from about 2 to 5 m$\bar\rho(z)$7 s$\bar\rho(z)$8, and passive-tracer mixing increases by about 15% near the northern tropopause in the range 10–15 km and 55–80°N (Banerjee et al., 28 Aug 2025).

MS-GWaM has also been used in ICON to examine the consequences of relaxing three “non-classical” assumptions simultaneously: pseudomomentum-only forcing, instantaneous adjustment, and one-dimensional propagation. In that study, transience and lateral propagation have the strongest influence on both the monthly mean zonal-mean circulation and on migrating and nonmigrating solar tides. The mean circulation in the mesosphere and lower thermosphere is affected at all latitudes, and the stratosphere is modified at high altitudes as well. In comparisons with SABER/TIMED tides, the 3D direct MS-GWaM configuration gives the most favorable agreement. For example, in December the observed DW1 temperature tide has a maximum near 95 km, whereas SS and 1D place it around 110 km; the 3D configuration moves the maximum closer to 100 km. The same study reports that the SW2 dual-maximum structure is reproduced more realistically in 3D than in SS or 1D configurations (Kühner et al., 26 Sep 2025).

A complementary research direction replaces explicit ray tracing with machine learning trained on resolved global fluxes. The global Attention U-Net study on the WINDSET dataset shares MS-GWaM’s central scientific motivation—moving beyond the single-column approximation toward three-dimensional, laterally propagating gravity-wave dynamics—but it does so by learning a global mapping

$\bar\rho(z)$9

rather than by explicitly evolving rays, wave action, or phase-space transport. The paper therefore positions the ML approach as complementary rather than duplicative relative to MS-GWaM (Gupta et al., 2024).

Taken together, these studies define MS-GWaM not as a single drag formula but as a multi-scale atmospheric wave framework: WKB-derived, spectral, prognostic, and phase-space based; capable of transient and oblique propagation; compatible with direct momentum and entropy forcing; and designed to bridge the gap between classical gravity-wave parameterizations and fully wave-resolving simulations (Achatz et al., 2023).