Internal Gravity Waves: Dynamics in Stratified Media

• Internal gravity waves are buoyancy-driven oscillations in stratified fluids that transport energy, momentum, and chemical tracers.
• They are generated by turbulent convective motions and wind shear, exhibiting dispersion governed by the Brunt–Väisälä frequency and anisotropic propagation.
• Nonlinear interactions, wave breaking, and MHD effects modulate their behavior, leading to phenomena such as energy trapping and topologically protected states.

Internal gravity waves (IGWs) are buoyancy-driven oscillations that propagate in stably stratified fluids, where the restoring force is gravity acting on density variations. Ubiquitous in geophysical and astrophysical contexts, IGWs play essential roles in mediating energy, momentum, and composition transport in the Earth's oceans and atmosphere, the Sun, and stellar interiors. Their behaviors, excitation mechanisms, and impacts depend sensitively on the properties of the medium—especially stratification, rotation, magnetic fields, and nonlinear feedbacks.

1. Fundamental Theory and Dispersion Properties

Internal gravity waves are characterized by their ability to propagate where the stratification is stable (Brunt–Väisälä frequency $N^2 > 0$). In the Boussinesq or anelastic approximation, the canonical IGW dispersion relation is

$\omega^2 = N^2\,\frac{k_h^2}{k_h^2 + k_z^2}$

where $\omega$ is the wave frequency, $k_h$ the horizontal, and $k_z$ the vertical wavenumber. The Brunt–Väisälä (buoyancy) frequency determines the maximum allowable frequency for IGWs. In compressible, stratified media, such as stellar interiors or the solar atmosphere, the dispersion relation generalizes and couples to acoustic modes (Vigeesh et al., 2020).

For $\omega \ll N$, IGWs propagate with phase velocities inclined to the horizontal, and the vertical group velocity is antiparallel to the phase velocity. The associated energy and momentum fluxes are proportional to products such as $\langle p' v_z\rangle$ (mechanical) or $\langle \rho v^2 v_z\rangle$ for kinetic energy transport (Vigeesh et al., 2020, Rogers et al., 2013).

2. Generation Mechanisms Across Environments

Solar and Stellar Interiors:

IGWs are mainly excited at convective–radiative boundaries via Reynolds stresses from turbulent eddies and penetrative plumes. In stellar models, both direct numerical simulations and theory demonstrate that convective overshoot generates a broad spectrum of IGWs, with energy flux scaling

$F_\mathrm{IGW} \sim F_\mathrm{conv}\,M$

where $F_\mathrm{conv}$ is the convective flux and $M$ the convective Mach number. For smooth radiative–convective transitions, the IGW flux can reach $F_\mathrm{IGW} \sim F_\mathrm{conv}\, (d/H)$, potentially far exceeding the sharp-interface result (Lecoanet et al., 2012). Convectively excited IGWs exhibit an $\omega^{-13/2}$ or steeper frequency spectrum, with the efficiency of excitation modulated by interface structure and mode coupling (Lecoanet et al., 2012, Rogers et al., 2013, Lecoanet et al., 2021).

Solar Atmosphere:

IGWs originate when granular upflows overshoot into the stably stratified photosphere and chromosphere, launching buoyancy waves that propagate vertically (Vigeesh et al., 2020, Vigeesh et al., 2019).

Atmosphere and Ocean of Earth:

At planetary scales, IGWs are excited by wind shear, orography, frontogenesis, and even by parametric resonance—the background stratification or temperature gradient can be modulated (naturally or by seismic precursors) to parametrically amplify IGW modes, often as precursors to major atmospheric or oceanic events (Chefranov et al., 2013). In the ionosphere, IGW excitation and propagation are strongly influenced by electromagnetic effects, specifically Pedersen conductivity, which damps and decorrelates vortex structures (Misra et al., 2022).

3. Propagation, Dissipation, and Nonlinear Dynamics

Linear Propagation and Damping:

In the absence of dissipation, IGWs are highly anisotropic, with energy confined to paths constrained by local $N(z)$. Thermal diffusion and, to a far lesser extent, viscosity, yield a radiative damping rate for amplitude

$$\sim \exp\left[-\frac{1}{2}\tau(r)\right], \quad \tau(r) = [\ell(\ell+1)]^{3/2}\int K \frac{N N_T^2}{\omega^4}\frac{dr}{r^3}$$

(Mathis, 21 Nov 2024, Rogers et al., 2013). Radiation zones of stars and chromospheric layers of the solar atmosphere can be effective sinks for IGW energy.

Nonlinear Interactions:

Wave–wave and wave–mean-flow interactions generate triad coupling (transfer between frequencies and wavenumbers), self-interaction (generation of higher harmonics), and breaking. Fully nonlinear simulations in both solar-like stars and the Earth's atmosphere show that IGWs can self-organize into coherent vortices, undergo modulational instability (MI), or even collapse under nonlocal wave–mean coupling (Lashkin et al., 2023, Lashkin et al., 22 Aug 2024, Adhikary et al., 1 Nov 2025). In the focusing nonlinear Schrödinger (NLS) regime, localized “rogue" IGW events, Akhmediev/Kuznetsov–Ma breathers, and dark/gray solitons can emerge, with the regime (focusing/defocusing) set by a combination of vertical and horizontal scales (Lashkin et al., 2023, Lashkin et al., 22 Aug 2024).

Critical Layers and Wave Breaking:

At radii where the Doppler-shifted wave frequency vanishes (critical layers), IGWs interact with local shear. If the Richardson number is large (locally stable), IGWs are strongly attenuated, depositing momentum. For sufficiently low Richardson number, over-reflection or transmission occurs, and the unstable shear can amplify IGWs beyond their incident amplitude (Alvan et al., 2013). Nonlinear breaking, via convective overturning or shear instability, saturates the wave amplitude and enforces an upper bound on momentum transport—this now admits robust parametrizations for stellar models (Mathis, 21 Nov 2024).

4. Magnetohydrodynamic (MHD) Effects and Topology

Magnetized Atmospheres and Interiors:

The propagation of IGWs in magnetized media fundamentally alters via coupling to slow magnetosonic or Alfvén waves. In the solar atmosphere, the dimensionless plasma-$\beta$ regime determines wave behavior:

• For horizontal fields (internetwork), IGWs propagate upward largely unaffected.
• For vertical fields (network), IGWs reflect or convert into downward-propagating slow modes at the $\beta=1$ surface; upward mechanical flux is suppressed (Vigeesh et al., 2020, Lecoanet et al., 2016, Vigeesh et al., 2019).

When IGWs encounter a magnetic region with field strength above a critical threshold,

$B_c \propto \frac{\omega^2 r}{N}$

they convert perfectly into upward-propagating slow waves without reflection or continued IGW propagation (Lecoanet et al., 2016). This mechanism underlies the suppression of mixed dipole modes observed in strongly magnetized red giants.

Wave Band Gaps and Topological Surface States:

In oceanic or planetary interiors with periodically stratified (crystalline) density profiles, IGWs display band gaps (forbidden frequency ranges) and topologically protected surface states. These gaps, induced by Bragg scattering, inhibit vertical energy transport, while topological states allow highly localized wave channels along boundaries—distinct from classical ducting, these are set by the Zak phase topology of bulk bands (Ghaemsaidi et al., 2021).

5. Observational Diagnostics and Impacts

Astrophysical Diagnostics:

Asteroseismology reveals both smooth (red-noise) and peaked (g-mode) features at the stellar surface traceable to IGWs (Lecoanet et al., 2021, Ratnasingam et al., 2020). The detailed spectrum—its slope, scaling with frequency, and spatial/temporal coherence—is a direct probe of excitation, propagation, amplitudes, and damping mechanisms and enables constraints on core convection and mixing.

Solar Atmosphere:

IGWs can be detected via Doppler shifts in spectral lines, with rigorous cross-spectrum techniques required to preserve phase coherence. Overestimation of energy flux is possible when line-formation heights are not suitably close; spectral diagnostics need to respect phase-coherence limitations (Δz ≲ 200 km in the mid-photosphere) to yield reliable fluxes (Vigeesh et al., 2019).

Earth’s Oceans and Atmosphere:

Detection via consistency relations (kinetic-to-potential energy ratios) is complicated by standing-wave effects in low vertical modes reflected from the ocean bottom. New, depth- and mode-resolved formulae are required for interpretation near boundaries (Wang et al., 5 Nov 2025). In the lower atmosphere, IGW-induced turbulence, spectral slopes (e.g., $k_x^{-1.67}$, $k_z^{-2.89}$ in the troposphere), and episodic focusing/bursting are confirmed by observations and simulations (Adhikary et al., 1 Nov 2025, Lashkin et al., 2023, Lashkin et al., 22 Aug 2024). Competing acoustic resonance modes in the troposphere–stratopause cavity can resemble IGWs but typically lack vertical phase tilts and do not transport momentum vertically (Kochin, 2022).

Stellar Evolution Impacts:

IGWs redistribute angular momentum on secular timescales, explaining discrepancies between observed and predicted stellar spin profiles. Both linear (via radiative damping and critical layers) and nonlinear (via breaking) torque prescriptions now exist for 1D models (Rogers et al., 2013, Mathis, 21 Nov 2024, Alvan et al., 2014, Alvan et al., 2013).

6. Chemical Mixing and Secular Diffusion

IGWs promote mixing of trace chemical species in stratified stellar radiation zones. 2D simulations reveal that particle transport is effectively diffusive, with local diffusivity scaling as the square of the wave amplitude,

$D(r) = \mathcal{A} v_\mathrm{wave}^2(r)$

where the amplitude $v_\mathrm{wave}(r)$ follows from a balance between excitation, stratification, and radiative damping. This parametrization, with minimal new free parameters, is suited for integration into 1D stellar evolution codes, linking wave mixing directly to observable abundance correlations and asteroseismic inferences (Rogers et al., 2017).

7. Instabilities, Turbulence, and Future Directions

Modulational Instability, Collapse, and Turbulence:

Envelope dynamics of IGWs under the NLS reduction admit both modulational instability and, in 2D, critical collapse for high enough wave intensity—even when the underlying nonlinearity is nonlocal due to wave–mean-flow coupling (Lashkin et al., 2023, Lashkin et al., 22 Aug 2024). These routes produce enhanced mixing, anisotropic turbulence (horizontal/vertical spectra), and isolated extreme events akin to rogue waves.

Thermoacoustic Coupling:

Nonlinear coupling of IGWs to thermal waves in stratified atmospheres gives rise to thermoacoustic instabilities, leading to dual-cascade turbulence and spectral signatures tied to the lapse rate and feedback strength. Spectral slopes in simulated turbulence match those observed in the troposphere, validating the physical model (Adhikary et al., 1 Nov 2025).

Open Issues and Prospects:

• The dynamical evolution of the IGW spectrum in complex magnetic and rotating environments remains an area of active research (Vigeesh et al., 2020, Mathis, 21 Nov 2024).
• The connection between nonlinear IGW events and atmospheric phenomena such as traveling ionospheric disturbances or episodic mixing in the stratosphere is being elucidated via increasingly detailed numerical and laboratory studies (Lashkin et al., 22 Aug 2024, Adhikary et al., 1 Nov 2025).
• The role of topologically protected edge states in layered natural systems, including their potential to localize and trap energy, is an emerging area in geophysical fluid dynamics (Ghaemsaidi et al., 2021).

The multifaceted dynamics of internal gravity waves—spanning linear propagation, mode conversion, nonlinearity, turbulence, and global-scale momentum and composition transport—position IGWs as an indispensable framework for understanding stratified fluids across planetary and astrophysical systems.