Atmospheric Gravity Waves (AGWs)

• Atmospheric gravity waves (AGWs) are buoyancy-driven disturbances in stratified atmospheres that enable vertical energy and momentum transfer.
• They encompass both high-frequency acoustic and low-frequency internal gravity branches, defined by distinct dispersion relations and polarization diagnostics.
• AGWs are detected via multisensor approaches on Earth, Mars, Venus, and the Sun, impacting weather forecasting, climate models, and planetary studies.

Atmospheric gravity waves (AGWs) are buoyancy-driven disturbances in stratified atmospheres that transport energy and momentum across widely separated height ranges; in part of the theoretical literature, the label also encompasses the broader acoustic-gravity wave system, in which a high-frequency acoustic branch and a low-frequency internal gravity branch are coupled by stratification, rotation, and background structure (Chatterjee et al., 2021, Cheremnykh et al., 2019). Across Earth, Mars, Venus, and the solar atmosphere, AGWs are treated as a primary mechanism of vertical coupling, a source of mesospheric, thermospheric, and chromospheric variability, and a diagnostic of winds, magnetic topology, and unresolved subgrid-scale fluxes (Yiğit et al., 2021, Silva et al., 2021, Vesa et al., 2023).

1. Definition, branches, and linear structure

In the linear theory summarized in the recent AGW literature, the governing distinction is between a high-frequency acoustic branch and a low-frequency internal gravity branch. With the Coriolis force retained, the general linear dispersion relation derived for a stratified atmospheric fluid is

$$\omega^4 - \omega^2\left(\omega_a^2 + c_s^2k^2 + 4\Omega_0^2\right) + c_s^2\omega_g^2 k_\perp^2 + 4\Omega_0^2(c_s^2k_z^2 + \omega_a^2) = 0,$$

with branch separation given by

$\omega_h^2 = \omega_a^2 + c_s^2 k^2$

for the high-frequency acoustic branch and

$$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$

for the low-frequency internal gravity branch. In that formulation, the Coriolis term increases the IGW frequency, while the high-frequency acoustic branch is unaffected by the Coriolis force (Chatterjee et al., 2021).

A complementary formulation treats AGWs as a two-frequency oscillatory process. In that approach, the disturbance field is represented in real variables, and the dynamics are organized around two eigenfrequencies,

$$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$

so that the perturbed velocities are sums of terms oscillating at both eigenfrequencies. This representation is used to account for beat phenomena, amplitude modulation, and in-phase oscillations of velocity components that are not naturally emphasized in single-frequency descriptions (Cheremnykh et al., 2019).

Polarization relations provide the operational bridge between theory and observation. For freely propagating AGWs, the temperature and vertical-velocity perturbations satisfy

$$\frac{T'}{T} = (\gamma - 1) \frac{k_z}{\omega} V_z,$$

while phase relations for density and temperature relative to vertical velocity are written as

$$\tan \Delta_{\rho, V_z} = -\frac{1 - 2(1 - \gamma^{-1})(C_s k_x / \omega)^2}{2(k_z H)},$$

$$\tan \Delta_{T, V_z} = \frac{1 - 2\gamma^{-1}(C_s k_x / \omega)^2}{2(k_z H)}.$$

These relations differ sharply from the evanescent case, where $\rho'/\rho$ and $T'/T$ are in-phase or anti-phase with each other but both are phase-shifted by $\pm \pi/2$ relative to $\omega_h^2 = \omega_a^2 + c_s^2 k^2$0. The resulting diagnostic diagram in $\omega_h^2 = \omega_a^2 + c_s^2 k^2$1 space distinguishes the gravity branch, the acoustic branch, freely propagating waves, and evanescent modes, and can also recover the sign of $\omega_h^2 = \omega_a^2 + c_s^2 k^2$2, hence the vertical propagation direction (Klymenko et al., 2021).

A recurrent source of ambiguity in the literature is terminological rather than physical: some studies reserve “gravity waves” for the buoyancy branch alone, while others use “acoustic-gravity waves” for the coupled system. The dispersion relations and polarization diagnostics show that the distinction is formalized at the level of branch structure rather than by incompatible physics. This suggests that AGW nomenclature is context-dependent, especially when acoustic cut-off effects, evanescence, or branch coupling are central to the analysis.

2. Propagation in realistic atmospheres

Away from the idealized adiabatic, isothermal limit, AGW propagation is altered by dissipation, non-isothermality, thermal relaxation, winds, and rotation. For a horizontally homogeneous isothermal atmosphere with heat conductivity and viscosity, the linear disturbance problem reduces to a sixth-order ordinary differential equation. In the nonviscous heat-conducting limit, the equation is reducible to a generalized hypergeometric form, enabling analytical study of attenuation, reflection, and the transition from quasi-classical to strongly dissipative behavior. A critical dissipation height $\omega_h^2 = \omega_a^2 + c_s^2 k^2$3 is defined by

$\omega_h^2 = \omega_a^2 + c_s^2 k^2$4

and is used to classify which waves can reach upper-atmospheric altitudes and contribute to middle-scale traveling ionospheric disturbances via waveguide leakage (Rudenko, 2013).

The real upper atmosphere also admits structures not present in the simplest textbook treatment. An overview of model development reports a previously unknown inelastic mode, a family of evanescent pseudo-modes, temperature-jump effects at the boundary of two isothermal half-spaces, modifications of acoustic and gravitational regions by vertical non-isothermality, additional damping associated with density gradients, and continuous spectra of evanescent AGWs modified by atmospheric rotation (Cheremnykh et al., 2023). These results broaden the admissible AGW state space beyond the standard freely propagating acoustic and gravity branches.

Non-adiabatic thermodynamics changes the cut-off structure directly. In an isothermal atmosphere with heating and cooling represented by power laws, the acoustic-gravity-wave dispersion relation acquires a thermal-relaxation term, and in regimes with an overwhelming influence of thermal processes the acoustic cut-off frequency decreases up to $\omega_h^2 = \omega_a^2 + c_s^2 k^2$5 times. The same analysis reports that the maximum frequency of the gravitational mode decreases with increasing power of thermal processes (Riashchikov et al., 2023).

Horizontally non-uniform wind flows introduce a separate amplitude control mechanism. For AGWs propagating through a medium with slowly varying wind speed, the derived amplitude law is

$\omega_h^2 = \omega_a^2 + c_s^2 k^2$6

where the linear factor reflects changes in background parameters and the exponential factor reflects inertia-force effects. The exponential term yields amplitude growth in headwind and amplitude decrease in downwind, and the theoretical dependence is reported to agree with satellite observations of AGWs in the polar thermosphere (Fedorenko et al., 2023).

A common oversimplification is to treat dissipation, thermal relaxation, and background flow as secondary corrections. The recent analytical literature indicates the opposite: these terms alter admissible spectra, cut-offs, attenuation rates, and even whether an observed disturbance can remain observable at upper-atmospheric heights.

3. Detection and diagnostic methodologies

Recent AGW diagnostics combine polarization analysis, spectral filtering, cross-correlation, radiowave remote sensing, and image-based pattern extraction. In satellite measurements, simultaneous fluctuations of velocity, density, temperature, and pressure can be used to infer wave type from phase shifts alone. Verification with Dynamics Explorer 2 showed that thermospheric polar disturbances mainly correspond to the gravity branch of freely propagating AGWs moving from bottom to top, and that no evanescent waves were observed on the considered orbits (Klymenko et al., 2021).

A distinct solar diagnostic is based on the three-dimensional cross-correlation function of vertical velocities measured at different heights. After filtering below the Lamb line and below the Brunt–Väisälä frequency, the inverse Fourier transform of the cross-spectrum,

$\omega_h^2 = \omega_a^2 + c_s^2 k^2$7

produces annular structures in spatial lag for upward-propagating IGW packets. The ring radius tracks horizontal group displacement, its width reflects the packet envelope, and imposed background flows shift the ring center in the flow direction. The same annular signature is reported in both synthetic data and multi-height Doppler observations (Calchetti et al., 2020).

Ground-based VLF radio sensing supplies a mesopause diagnostic. Along the GQD–A118 radio path, evening terminator forcing produced a systematic increase in fluctuations of radio-signal amplitudes, with dominant AGW periods in the 15–20 minute range. The inferred AGW amplitudes were 12–14\% in relative concentration fluctuations, corresponding to vertical displacements of 1.1–1.3 km, and energy-balance analysis indicated quasi-horizontal propagation at the terminator (O. et al., 2023).

Visible imaging from the International Space Station shows that even small-scale AGWs can be recovered from low-cost hardware when spatial resolution is sufficient. Using a Raspberry Pi camera module and post-flight 2D FFT analysis of cloud scenes, horizontal wavelengths from 1.0 to 4.7 km were identified in different northern-hemisphere regions (Magalhães et al., 2021).

These methodologies are not interchangeable. Polarization relations identify mode type and propagation direction; cross-correlation emphasizes group transport; VLF diagnostics infer mesospheric perturbations through reflection-height variability; and image-based Fourier methods recover horizontal scales in cloud tracers. A plausible implication is that robust AGW characterization increasingly depends on multimodal rather than single-instrument inference.

4. Planetary atmospheres

On Mars, thermospheric gravity-wave activity has been characterized directly from MAVEN/NGIMS measurements during a recent solar minimum. Using relative CO$\omega_h^2 = \omega_a^2 + c_s^2 k^2$8 density fluctuations $\omega_h^2 = \omega_a^2 + c_s^2 k^2$9, with $$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$0 obtained from a 7th-order polynomial fit to the logarithm of density along each orbit, a 10-month climatology over approximately 1300 orbits was constructed for 150–230 km. Monthly mean GW activity varies between 6–25\%, reaches a maximum at $$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$1 km, decays above 180–190 km, and remains 5–15\% near 230 km. Nighttime values exceed daytime values at all altitudes and seasons, activity increases with solar zenith angle, and middle-to-high latitudes show stronger activity than the lower atmosphere does (Yiğit et al., 2021).

The same Martian study interprets the day-night contrast through competition between wave growth due to decreasing background density and damping by molecular diffusion. The analysis concludes that convective instability is likely to play a less important role in limiting GW amplitudes in the upper thermosphere, and that molecular diffusion controls the amplitude maximum and its local-time dependence (Yiğit et al., 2021). This is one of the clearer recent examples of an observationally tested controversy in AGW physics.

A lower-altitude Martian application concerns CO$$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$2 ice clouds. In the MPI-MGCM with a whole-atmosphere subgrid-scale gravity-wave parameterization, unresolved harmonics are propagated through a transmissivity factor,

$$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$3

and cloud formation is diagnosed probabilistically through

$$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$4

In that framework, GWs cool the upper atmosphere of Mars by $$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$5, facilitate high-altitude CO$$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$6 cloud formation, and are described as a necessary physical mechanism for CO$$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$7 ice cloud formation in the Martian upper atmosphere during all seasons (Yiğit et al., 2018).

On Venus, systematic analysis of nightside lower-cloud imagery from VIRTIS-M and Akatsuki IR2 characterized almost 300 wave packets across more than 5500 images. Mean horizontal wavelengths were 100 km for VIRTIS and 158 km for IR2, packet widths were 250 km and 527 km, packet lengths were 486 km and 716 km, and most vertical wavelengths were 2–5 km. Phase velocities ranged up to $$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$8 m/s, and most packets exhibited a 2–6\% optical-thickness drop between crest and trough. The waves were interpreted as internal gravity waves, with convective instability in lower or middle clouds identified as a likely source and topographic forcing treated as possible but not dominant (Silva et al., 2021).

Taken together, the planetary results show AGWs acting in markedly different dynamical roles: thermospheric variability on Mars, cloud microphysical preconditioning on Mars, and lower-cloud mesoscale structure on Venus. This suggests that cross-planet comparison is most informative when framed in terms of wave-mediated coupling rather than a single canonical altitude range.

5. Solar atmosphere

In the solar atmosphere, AGWs are treated as low-frequency waves generated by turbulent convection and propagating obliquely through the photosphere and chromosphere. Multi-height observations near quiet-Sun disk center, using IBIS together with SDO/HMI and SDO/AIA, identify propagating AGWs through negative phase differences at low frequency and low horizontal wavenumber. Time-distance analysis yields an average group speed of $$\omega_l^2 = \frac{k_\perp^2\omega_g^2}{k_z^2 + 1/4H^2} + 4\Omega_0^2$$9 km s$$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$0, and the observed median magnetic field of 4.2 G suggests that propagating AGWs are not significantly affected by quiet-Sun photospheric magnetic fields in that regime (Vesa et al., 2023).

A separate spectroscopic study with H$$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$1, Ca II IR, and Fe I 6173 Å reports that AGWs are detected within or near magnetic flux concentration regions, where spicules are also predominant, and that power increases with height in the chromosphere. Those regions also feature inclined magnetic fields, which might be contributing to the propagation of these low-frequency AGWs, and average power maps at spicule locations reveal significant power at AGW frequency across different chromospheric heights (Chaurasiya et al., 9 Jan 2025).

Magnetic modulation becomes clearer outside the quiet-Sun weak-field limit. High-resolution multi-wavelength observations and CO$$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$2BOLD simulations show that AGWs are efficiently suppressed and/or reflected in intermediate to strong, vertically oriented fields in the upper photosphere, while they propagate rather freely in QS and transverse fields. The observational signature is strongest in spatial coherence-weighted phase-difference maps and binned coherence-weighted phase differences versus field strength and inclination; direct signatures in observed $$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$3 phase maps are reported to be less discernible, likely because of spatial averaging and height-of-formation ambiguities (Vesa et al., 24 Sep 2025).

Thermal physics modifies the solar AGW spectrum as well as magnetic topology does. In an isothermal atmosphere with thermal misbalance, the AGW dispersion relation contains an explicit relaxation term,

$$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$4

and the acoustic cut-off frequency is reduced in the strong-thermal-process regime (Riashchikov et al., 2023).

The solar literature therefore supports a regime-dependent interpretation of magnetic influence. Weak quiet-Sun photospheric fields do not significantly alter the observed AGW propagation in one set of observations, whereas intermediate-to-strong vertical fields suppress or reflect AGWs in the upper photosphere in another. This suggests that field strength, inclination, and diagnostic height must be treated jointly rather than as interchangeable “magnetic effects.”

6. Modeling, parameterization, and data-driven analysis

AGWs remain a central subgrid-scale problem in atmosphere and climate models. For wind-energy LES under linearly stratified conditions, internal-wave properties are controlled mainly by the Froude number, and the recommended numerical setup scales with effective wavelengths and the Brunt–Väisälä frequency. The horizontal domain length should accommodate at least one effective horizontal wavelength, the vertical domain height and damping-layer thickness at least one effective vertical wavelength, and the Rayleigh damping parameter $$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$5 should lie in the range $$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$6. When those recommendations are followed, reflection coefficients remain below $$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$7 (Khan et al., 2024).

At the circulation-model level, the transient IDEMIX model parameterizes nonorographic internal gravity waves through prognostic energy densities in multiple azimuthal compartments, derived from an integrated full wave spectrum. Its compartment equations evolve directionally resolved energies such as $$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$8 and $$\omega_{1,2}^2 = \frac{1}{2} \left(\omega_{01}^2 + \omega_{02}^2 \pm \sqrt{D}\right),$$9, while saturation maintains the wave field within convective stability limits and a critical-layer closure controls upward flux penetration. IDEMIX reproduces zonal gravity-wave drag around the mesopause, shows a reversal of gravity-wave drag around the mesopause region due to changes in momentum flux there, and captures the summer hemisphere flow reversal, the cold summer mesospheric pole, and alternating positive and negative structures in the meridional mean flow (Quinn et al., 2023).

Machine-learning approaches are now used for both detection and parameterization. For noisy satellite imagery, gWaveNet integrates a checkerboard custom kernel into the first convolutional layer of a 15-layer CNN and reports over 98\% training accuracy, over 96\% validation accuracy, over 94\% test accuracy, and F1 score $$\frac{T'}{T} = (\gamma - 1) \frac{k_z}{\omega} V_z,$$0, with the trainable $$\frac{T'}{T} = (\gamma - 1) \frac{k_z}{\omega} V_z,$$1 kernel giving the best reported performance (Mostafa et al., 2024). For subgrid-scale parameterization, fine-tuning a pre-trained encoder-decoder from the 2.3 billion parameter Prithvi WxC foundation model yields a gravity-wave parameterization with Hellinger distance 0.06 versus 0.11 for an Attention U-Net baseline, while also improving generalization to regions and levels excluded from pre-training (Gupta et al., 4 Sep 2025).

The broader significance of these model developments is methodological as much as physical. Traditional parameterizations are described as highly simplified and poorly constrained by observations, while newer transient spectral schemes and fine-tuned foundation models attempt to retain intermittency, horizontal propagation, and evolving momentum flux. A plausible implication is that future AGW research will increasingly couple physically based closures, high-resolution reanalysis, and task-specific machine learning rather than treating parameterization and detection as separate problems.