---
title: 'MS-GWaM: Multi-Scale Gravity-Wave Model'
url: https://www.emergentmind.com/topics/multi-scale-gravity-wave-model-ms-gwam
type: topic
---

# MS-GWaM: Multi-Scale Gravity-Wave Model

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 [2310.07334] [2309.11257] [2408.03139].

## 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 [2310.07334].

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 [2310.07334].

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 [2310.07334].

## 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 \(N(X,k,T)\) or \(\mathcal N(\mathbf{x},t,\mathbf{k})\) 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
\[
\left(\partial_T+c\cdot\nabla_X\right)\omega=\partial_T\Omega, \qquad
\left(\partial_T+c\cdot\nabla_X\right)k=-\nabla_X\Omega,
\]
with group velocity
\[
c=\nabla_k\Omega.
\]
For the spectral density, the core transport law is
\[
\partial_T N + \nabla_X\cdot(cN)+\nabla_k\cdot(\dot{k}N)=0,
\]
or, because the phase-space velocity is non-divergent, equivalently
\[
\partial_T N + c\cdot\nabla_X N + \dot{k}\cdot\nabla_k N = 0.
\]
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** [2310.07334] [2408.03139].

Wave-action conservation is inherited from the underlying WKB theory. In the locally monochromatic setting,
\[
\partial_T \mathcal A + \nabla_X\cdot(c\mathcal A)=0, \qquad
\mathcal A=\frac{E_{gw}}{\omega}.
\]
The theory also includes explicit conservation statements for **energy** and an extended **potential-vorticity-like invariant**,
\[
\Pi = P - e\cdot\nabla\times\frac{p}{\rho},
\]
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 [2310.07334].

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 [2509.21939].

## 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 \((x,k)\)-space according to
\[
\frac{d}{dt}(x,k)=(c,\dot{k}),
\]
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 [2310.07334] [2309.11257].

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 [2310.07334]. In the idealized orographic application, the sink is written as
\[
\mathcal S_0=-2D|\mathbf{k}|^2\mathcal N,
\]
which acts as a turbulent-diffusion representation of saturation and breaking [2408.03139].

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 [2309.11257] |
| **MS-GWaM-3D** | Full 3D transient propagation | Global non-orographic waves in ICON [2309.11257] |
| **WKB-TR** | Fully transient WKB/MS-GWaM | Idealized orographic tests in PincFlow [2408.03139] |
| **WKB-ST** | Steady-state WKB/MS-GWaM | Classic mountain-wave comparator [2408.03139] |

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 \(\varepsilon_q = 7\%\) [2309.11257]. By contrast, the idealized orographic experiments are conducted in **PincFlow**, with two-dimensional, isothermal flow over a **monochromatic lower boundary**
\[
h(x)=\frac{h_0}{2}\left[1+\cos\!\left(\frac{\pi x}{l_0}\right)\right],
\]
mountain heights from \(h_0=100\) m to \(h_0=1000\) m, a rigid upper boundary, a Rayleigh damping layer aloft, and a linear three-hour spin-up of the orography [2408.03139].

## 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 [2310.07334] [2509.21939].

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
\[
\omega_\alpha = -\mathbf{k}_\alpha\cdot \mathbf{u}_m,
\]
and wave action is simply set to zero above levels where \(\omega \to 0\), 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 [2408.03139].

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 [2309.11257].

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** [2408.03139].

## 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 [2408.03139].

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 [2408.03139].

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 [2408.03139].

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 [2408.03139].

## 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 [2309.11257].

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\(^2\) s\(^{-2}\)**, and passive-tracer mixing increases by about **15%** near the northern tropopause in the range **10–15 km** and **55–80°N** [2508.20562].

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 [2509.21939].

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
\[
(u,v,\theta)\mapsto (u'w',v'w')
\]
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 [2406.14775].

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 [2310.07334].

Source: https://www.emergentmind.com/topics/multi-scale-gravity-wave-model-ms-gwam